$\aleph _0$-categoricity of semigroups II

Abstract

A countable semigroup is $\aleph _0$-categorical if it can be characterised, up to isomorphism, by its first-order properties. In this paper we continue our investigation into the $\aleph _0$-categoricity of semigroups. Our main results are a complete classification of $\aleph _0$-categorical orthodox completely 0-simple semigroups, and descriptions of the $\aleph _0$-categorical members of certain classes of strong semilattices of semigroups.

1 Introduction

A countable structure is $\aleph _0$-categorical if it is uniquely determined by its first-order properties, up to isomorphism. While the concept of $\aleph _0$-categoricity arises naturally from model theory, it has a purely algebraic formulation thanks to the Ryll-Nardzewski theorem (RNT). Independently accredited to Engeler [3], Ryll-Nardzewski [28] and Svenonius [29], it states that the $\aleph _0$-categoricity of a structure M is equivalent to there being only finitely many orbits in the natural action of Aut(M) (the automorphism group of M) on $M^n$, for each $n\ge 1$. Significant results exist for both relational and algebraic structures from the point of view of $\aleph _0$-categoricity, but, until recently, little was known in the context of semigroups. This article is the second of a pair initiating and developing the study of $\aleph _0$-categorical semigroups. For background and motivation we refer the reader to Hodges [14] and Evans [5], and to our first article [9].

We explore in [9] the behaviour of $\aleph _0$-categoricity with respect to standard constructions, such as quotients and subsemigroups. For example, $\aleph _0$-categoricity of a semigroup is inherited by both its maximal subgroups and its principal factors. Differences with the known theory for groups and rings emerged, for example, any $\aleph _0$-categorical nil ring is nilpotent, but the same is not true for semigroups. While keeping the machinery at a low level, we were able to give, amongst other results, complete classifications of $\aleph _0$-categorical primitive inverse semigroups and of E-unitary inverse semigroups with finite semilattices of idempotents.

For the work in this current article, it is helpful to develop some general strategies and then apply them in various contexts. In view of this, in Sect. 2, we introduce $\aleph _0$-categoricity in the setting of (first-order) structures. Although we will mostly be working in the context of semigroups, this broader view will be useful for studying structures, such as graphs and semilattices, which naturally arise in our considerations of semigroups. Key results from Gould and Quinn-Gregson [9] are given in this setting. In particular, we formalise the previously defined concept of $\aleph _0$-categoricity over a set of subsets; the $\aleph _0$-categoricity of rectangular bands over any set of subrectangular bands acts as a useful example.

In Sect. 3 we construct a handy method for dealing with the $\aleph _0$-categoricity of semigroups in which their automorphisms can be built from certain ingredients. This is then used in Sect. 4 to study the $\aleph _0$-categoricity of strong semilattices of semigroups. The main results of this article are in Sect. 5, where we continue from [9] our study into the $\aleph _0$-categoricity of completely 0-simple semigroups. We follow a method of Graham and Houghton by considering graphs arising from Rees matrix semigroups, which necessitated our study of $\aleph _0$-categoricity in the general setting of structures.

We assume that all structures considered will be of countable cardinality.

2 The $\aleph _0$-categoricity of a structure

We begin by translating a number of results in [9] to the general setting of (first-order) structures. Their proofs easily generalize, and as such we shall omit them, referencing only the corresonding result in [9].

A (first-order) structure is a set M together with a collection of constants ${{\mathfrak {C}}}$, finitary relations ${{\mathfrak {R}}}$, and finitary functions ${{\mathfrak {F}}}$ defined on M. We denote the structure as $(M;{{\mathfrak {R}}},{{\mathfrak {F}}},{{\mathfrak {C}}})$, or simply M where no confusion may arise. Each constant element is associated with a constant symbol, each n-ary relation is associated with an n-ary relational symbol, and each n-ary function is associated with an n-ary function symbol. The collection L of these symbols is called the signature of M. We follow the usual convention of not distinguishing between the constants/relations/functions of M, and their corresponding abstract symbols in L.

Our main example is that of a semigroup $(S,\cdot )$, where S is a set together with a single (associative) binary operation $\cdot \,$, and so the associated signature consists of a single binary function symbol.

A property of a structure is first-order if it can be formulated within first-order predicate calculus. A (countable) structure is $\aleph _0$-categorical if it can be uniquely classified by its first-order properties, up to isomorphism.

The central result in the study of $\aleph _0$-categorical structures is the Ryll-Nardzewski Theorem, which translates the concept to the study of oligomorphic automorphism groups (see [14]). Before stating it, it is worth fixing some notation and definitions. Let $\phi :A\rightarrow B$ be a map, let ${{\overline{a}}}=(a_1,\ldots ,a_n)$ be an n-tuple of A and let $M\subseteq A$. Then we let ${{\overline{a}}}\phi$ denote the n-tuple of B given by $(a_1\phi ,\ldots ,a_n\phi )$, and $M\phi$ denotes the subset $\{m\phi : m\in M\}$ of B.

Given a structure M, we say that a pair of n-tuples ${{\overline{a}}}=(a_1, \dots , a_n)$ and ${{\overline{b}}}=(b_1, \dots ,b_n)$ of M are automorphically equivalent or belong to the same n-automorphism type if there exists an automorphism $\phi$ of M such that ${{\overline{a}}}\phi ={{\overline{b}}}$, that is, $a_i\phi = b_i$ for each $i \in \{ 1,\ldots , n\}$. We denote this equivalence relation as ${{\overline{a}}} \, \sim _{M,n} \, {{\overline{b}}}$. We call Aut(M) oligomorphic if Aut(M) has only finitely many orbits in its action on $M^n$ for each $n\ge 1$, that is, if each $|M^n/\sim _{M,n}|$ is finite.

Theorem 2.1

(The Ryll-Nardzewski theorem (RNT)) A structure M is $\aleph _0$-categorical if and only if Aut(M) is oligomorphic.

It follows from the RNT that every $\aleph _0$-categorical structure is uniformly locally finite [14, Corollary 7.3.2], that is, there is a finite uniform bound on the size of the n-generated substructures, for each $n\ge 1$. In particular, an $\aleph _0$-categorical semigroup is periodic, with bounded index and period.

Another immediate consequence of the RNT is that any characteristic substructure inherits $\aleph _0$-categoricity, where a subset/substructure is called characteristic if it is invariant under automorphisms of the structure. However, key subsemigroups of a semigroup such as maximal subgroups and principal ideals are not necessarily characteristic, and a more general definition is required:

Definition 2.2

Let M be a structure and, for some fixed $t\in {{\mathbb {N}}}$, let $\{{{\overline{X}}}_i:i\in I\}$ be a collection of t-tuples of M. Let $\{A_i:i\in I\}$ be a collection of subsets of M with the property that for any automorphism $\phi$ of M such that there exists $i,j\in I$ with ${{\overline{X}}}_i \phi = {{\overline{X}}}_j$, then $\phi |_{A_i}$ is a bijection from $A_i$ onto $A_j$. Then we call ${\mathcal {A}}=\{(A_i,{{\overline{X}}}_i):i\in I\}$ a system of t-pivoted pairwise relatively characteristic (t-pivoted p.r.c.) subsets (or, substructure, if each $A_i$ is a substructure) of M. The t-tuple ${{\overline{X}}}_i$ is called the pivot of $A_i$ ($i\in I$). If $|I|=1$ then, letting $A_1=A$ and ${{\overline{X}}}_1={{\overline{X}}}$, we write $\{(A, X)\}$ simply as (A, X), and call A an ${{\overline{X}}}$-pivoted relatively characteristic (${{\overline{X}}}$-pivoted r.c.) subset/substructure of M.

In [9], Definition 2.2 was shown to be of use in regard to, for example, Green’s relations. In particular, $\{(H_e,e):e\in E(S)\}$ forms a system of 1-pivoted p.r.c. subgroups of a semigroup S. It then followed from the proposition below that maximal subgroups inherit $\aleph _0$-categoricity, and moreover there exists only finitely many non-isomorphic maximal subgroups in an $\aleph _0$-categorical semigroup.

Proposition 2.3

[9, Proposition 3.3] Let M be an $\aleph _0$-categorical structure and $\{(A_i,{{\overline{X}}}_i):i\in I\}$ a system of t-pivoted p.r.c. subsets of M. Then $\{|A_i| : i \in I\}$ is finite. If, further, each $A_i$ forms a substructure of M, then $\{A_i: i \in I\}$ is finite, up to isomorphism, with each $A_i$ $\aleph _0$-categorical.

We use the RNT in conjunction with [9, Lemma 2.8] to prove that a structure M is $\aleph _0$-categorical in the following way. For each $n\in {{\mathbb {N}}}$, let $\gamma _1,\ldots ,\gamma _r$ be a finite list of equivalence relations on $M^n$ such that $M^n/\gamma _i$ is finite for each $1\le i \le r$ and

$$\begin{aligned} \gamma _1\cap \gamma _2 \cap \cdots \cap \gamma _r \subseteq \, \sim _{M,n}. \end{aligned}$$

A consequence of the two aforementioned results is that M is $\aleph _0$-categorical. This result will often be drawn upon in a less formal way as follows. Suppose that we have an equivalence relation $\sigma$ on $M^n$ that arises from different ways in which a given condition may be fulfilled; if $M^n/\sigma$ is finite, then we say the condition has finitely many choices.

Example 2.4

Recalling [9, Example 2.10], consider the equivalence $\natural _{X,n}$ on n-tuples of a set X given by

$$\begin{aligned} (a_1,\ldots ,a_n) \, \natural _{X,n} \, (b_1,\ldots ,b_n) \text { if and only if } [a_i=a_j \Leftrightarrow b_i=b_j, \text { for each } i,j]. \end{aligned}$$

(2.1)

A pair of n-tuples ${{\overline{a}}}$ and ${{\overline{b}}}$ are $\natural _{X,n}$-equivalent if and only if there exists a bijection $\phi :\{a_1,\ldots ,a_n\} \rightarrow \{b_1,\ldots ,b_n\}$ such that $a_i\phi =b_i$, and the number of $\natural _{X,n}$-classes of $X^n$ is finite, for each $n\in {{\mathbb {N}}}$. Note also that if M is a structure then any pair of n-automorphically equivalent tuples are clearly $\natural _{M,n}$-equivalent.

Let M be a structure and ${\mathcal {A}}=\{A_i:i\in I\}$ a collection of subsets of M. We may extend the signature of M to include the unary relations $A_i$ ($i\in I$). We denote the resulting structure as ${{\underline{M}}}=(M;{\mathcal {A}})$, which we call a set extension of M. If ${\mathcal {A}}=\{A_1,\ldots ,A_n\}$ is finite, then we may simply write ${{\underline{M}}}$ as $(M;A_1,\ldots ,A_n)$.

Notice that automorphisms of ${{\underline{M}}}$ are simply those automorphisms of M which fix each $A_i$ setwise, that is automorphisms $\phi$ such that $A_i\phi =A_i$ ($i\in I$). The set of all such automorphisms will be denoted Aut($M;{\mathcal {A}}$), and clearly forms a subgroup of Aut(M). The $\aleph _0$-categoricity of ${{\underline{M}}}$ is therefore equivalent to our previous notion of M being $\aleph _0$-categorical over ${\mathcal {A}}$ in [9].

Lemma 2.5

[9, Lemma 5.2] Let M be a structure with a system of t-pivoted p.r.c. subsets $\{(A_i,{{\overline{X}}}_i):i\in I\}$. Then $(M;\{A_i:i\in I\})$ is $\aleph _0$-categorical if and only if M is $\aleph _0$-categorical and I is finite.

Lemma 2.6

[9, Lemma 5.3] Let M be a structure, let $t,r\in {{\mathbb {N}}}$, and for each $k\in \{1,\ldots ,r\}$ let ${{\overline{X}}}_k\in M^t$. Suppose also that $A_k$ is an ${{\overline{X}}}_k$-pivoted relatively characteristic subset of M for $1\le k \le r$. Then $(M;A_1,\ldots ,A_r)$ is $\aleph _0$-categorical if and only if M is $\aleph _0$-categorical.

Consequently, if S is an $\aleph _0$-categorical semigroup and $G_1,\ldots , G_n$ is a collection of maximal subgroups of S then $(S;G_1,\ldots ,G_n)$ is $\aleph _0$-categorical.

However, note that not every $\aleph _0$-categorical set extension of a semigroup requires the subsets to be relatively characteristic. We claim that any set extension of a rectangular band by a finite set of subrectangular bands is $\aleph _0$-categorical. This result is of particular use in the next section when considering the $\aleph _0$-categoricity of normal bands.

Recall that every rectangular band can be written as a direct product of a left zero and right zero semigroup. The following isomorphism theorem for rectangular bands will be vital for proving our claim, and follows immediately from Howie [17, Corollary 4.4.3]:

Lemma 2.7

Let $B_1=L_1\times R_1$ and $B_2=L_2\times R_2$ be a pair of rectangular bands. If $\phi _L:L_1\rightarrow L_2$ and $\phi _R:R_1\times R_2$ are a pair of bijections, then the map $\phi :B_1\rightarrow B_2$ given by $(l,r)\phi =(l\phi _L,r\phi _R)$ is an isomorphism, denoted $\phi =\phi _L\times \phi _R$ . Conversely, every isomorphism can be constructed this way.

Theorem 2.8

If B is a rectangular band and $B_1,\ldots ,B_r$ is a finite list of subrectangular bands of B, then ${{\underline{B}}}=(B;B_1,\ldots ,B_r)$ is $\aleph _0$-categorical. In particular, a rectangular band is $\aleph _0$-categorical.

Proof

Let $B=L\times R$, where L is a left zero semigroup and R is a right zero semigroup. For each $1\le k \le r$, let $L_k\subseteq L$ and $R_k\subseteq R$ be such that $B_k=L_k\times R_k$. Define a pair of equivalence relations $\sigma _L$ and $\sigma _R$ on L and R, respectively, by

$$\begin{aligned}&i \, \sigma _L \, j \Leftrightarrow [i\in L_k \Leftrightarrow j\in L_k, \text { for each } k], \\&i \, \sigma _R \, j \Leftrightarrow [i\in R_k \Leftrightarrow j\in R_k, \text { for each } k]. \end{aligned}$$

The equivalence classes of $\sigma _L$ are simply the set $L{\setminus } \bigcup _{1\le k \le r} L_k$ together with certain intersections of the sets $L_k$. Since r is finite, it follows that $L/\sigma _L$ is finite, and similarly $R/\sigma _R$ is finite. Let ${{\overline{a}}}=((i_1,j_1),\ldots ,(i_n,j_n))$ and ${{\overline{b}}}=((k_1,\ell _1),\ldots ,(k_n,\ell _n))$ be a pair of n-tuples of B under the four conditions that

• (1) $i_s \, \sigma _L \, k_s$ for each $1\le s \le n$,

• (2) $j_s \, \sigma _R \, \ell _s$ for each $1\le s \le n$,

• (3) $(i_1,\ldots ,i_n) \, \natural _{L,n} \, (k_1,\ldots ,k_n)$,

• (4) $(j_1,\ldots ,j_n) \, \natural _{R,n} \, (\ell _1,\ldots ,\ell _n)$,

where $\natural _{L,n}$ and $\natural _{R,n}$ are the equivalence relations given by (2.1). By conditions (3) and (4), there exists bijections

$$\begin{aligned} \phi _L:\{i_1,\ldots ,i_n\}\rightarrow \{k_1,\ldots ,k_n\} \text { and } \phi _R:\{j_1,\ldots ,j_n\}\rightarrow \{\ell _1,\ldots ,\ell _n\} \end{aligned}$$

given by $i_s\phi _L=k_s$ and $j_s\phi _R=\ell _s$ for each $1\le s\le n$. By condition (1), we can pick a bijection $\varPhi _L$ of L which extends $\phi _L$ and fixes each $\sigma _L$-classes setwise, and similarly construct $\varPhi _R$. Then $\varPhi =\varPhi _L\times \varPhi _R$ is an automorphism of B. Moreover, if $(i,j)\in B_k$ then $i\in L_k$ and as $i \, \sigma _L \, (i\varPhi _L)$ we have $i\varPhi _L\in L_k$. Dually, $j\in R_k$ and as $j \, \sigma _R \, (j\varPhi _R)$ we have $j\varPhi _R\in R_k$. Hence there exists $\ell \in L$ and $r\in R$ such that $(i\varPhi _L,r)$ and $(\ell ,j\varPhi _R)$ are in $B_k$, so that

$$\begin{aligned} (i\varPhi _L,r) (\ell ,j\varPhi _R) = (i\varPhi _L,j\varPhi _R)\in B_k \end{aligned}$$

as $B_k$ is a subrectangular band. We have thus shown that $(i,j)\varPhi =(i\varPhi _L,j\varPhi _R)\in B_k$, and so $B_k\varPhi \subseteq B_k$. We observe that $\varPhi ^{-1}=\varPhi ^{-1}_L\times \varPhi ^{-1}_R$ is also an automorphism of B with $\varPhi ^{-1}_L$ and $\varPhi ^{-1}_R$ setwise fixing the $\sigma _L$-classes and $\sigma _R$-classes, respectively. Following our previous argument we have $B_k\varPhi ^{-1}\subseteq B_k$, and so $B_k\varPhi =B_k$ for each k. Thus $\varPhi$ is an automorphism of ${{\underline{B}}}$, and is such that

$$\begin{aligned} (i_s,j_s)\varPhi =(i_s\varPhi _L,j_s\varPhi _R)=(i_s\phi _L,j_s\phi _R)=(k_s,\ell _s) \end{aligned}$$

for each $1\le s \le n$, so that ${{\overline{a}}} \, \sim _{{{\underline{B}}},n} \, {{\overline{b}}}$. Hence, as each of the four conditions on ${{\overline{a}}}$ and ${{\overline{b}}}$ have finitely many choices, it follows that ${{\underline{B}}}$ is $\aleph _0$-categorical. $\square$

Note that any set can be considered as a structure with no relations, functions or constants. Every bijection of the set is therefore an automorphism, and as such all sets are easily shown to be $\aleph _0$-categorical. In fact a simplification of the proof of Theorem 2.8 gives:

Corollary 2.9

Let M be a set, and $M_1,\ldots ,M_r$ be a finite list of subsets of M. Then $(M;M_1,\ldots ,M_r)$ is $\aleph _0$-categorical.

3 A new method: $(M,M';{{\underline{N}}};\varPsi )$-systems

For many of the structures we will consider, automorphisms can be built from isomorphisms between their components. For example, for a strong semilattice of semigroups $S=[Y;S_{\alpha };\psi _{\alpha ,\beta }]$, we can construct automorphisms of S from certain isomorphisms between the semigroups $S_{\alpha }$. In this example we also require an automorphism of the semilattice Y, which acts as an indexing set for the semigroups $S_{\alpha }$. We now extend this idea by setting up some formal machinery to deal with structures in which the automorphisms are built from a collection of data.

Notation 3.1

Given a pair of structures M and $M'$, we let Iso$(M;M')$ denote the set of all isomorphisms from M onto $M'$.

Definition 3.2

Let M be an L-structure with fixed substructure $M'$. Let ${\mathcal {A}} = \{M_i:i\in N\}$ be a set of substructures of $M'$ indexed by some K-structure N such that $M'=\bigcup _{i\in N} M_i$. Let $N_1,\ldots ,N_r$ be a finite partition of N, and set ${{\underline{N}}}=(N;N_1,\ldots ,N_r)$. For each $i,j\in N$, let $\varPsi _{i,j}$ be a subset of $\text {Iso}(M_i;M_j)$ under the conditions that

• (3.1) if $i,j\in N_k$ for some $1\le k\le r$ then $\varPsi _{i,j} \ne \emptyset$,

• (3.2) if $\phi \in \varPsi _{i,j}$ and $\phi '\in \varPsi _{j,\ell }$ then $\phi \phi '\in \varPsi _{i,\ell }$,

• (3.3) if $\phi \in \varPsi _{i,j}$ then $\phi ^{-1}\in \varPsi _{j,i}$,

• (3.4) if $\pi \in \text {Aut}({{\underline{N}}})$ and $\phi _i\in \varPsi _{i,i\pi }$ for each $i\in N$, then there exists an automorphism of M extending the $\phi _i$.

Letting $\varPsi =\bigcup _{i,j\in N} \varPsi _{i,j}$, then, under the conditions above, we call ${\mathcal {A}}=\{M_i:i\in N\}$ an $(M,M';{{\underline{N}}};\varPsi )$-system (in M). If $M'=M$ then we may simply refer to this as an $(M;{{\underline{N}}};\varPsi )$-system.

By Condition (3.1) if $i,j\in N_k$ for some k, then $M_i\cong M_j$. Hence the number of isomorphism types in ${\mathcal {A}}$ is bounded by r. Moreover, it follows from Conditions (3.1)–(3.3) that $\varPsi _{i,i}$ is a subgroup of $\text {Aut}(M_i)$, for each $i\in N$. If the sets $M_i$ are not pairwise disjoint, then Condition (3.4) should be met with caution. Indeed, if $x\in M_i\cap M_j$ then by taking $\pi$ to be the identity map of ${{\underline{N}}}$, we have that $x\phi _i,x\phi _j\in M_i\cap M_j$ for all $\phi _i\in \text {Aut}(M_i)$ and $\phi _j\in \text {Aut}(M_j)$ by Condition (3.4). However, for our work the sets $M_i$ will mostly be pairwise disjoint, or will all intersect at an element which is fixed by every isomorphism between the $M_i$. For example, M could be a semigroup containing a zero, and 0 is the intersection of each of the sets $M_i$.

Note also that no link needs to exist between the signatures L and K. For most of our examples they will be the signature of semigroups and the signature of sets (the empty signature), respectively.

Given an $(M;M';{{\underline{N}}};\varPsi )$-system ${\mathcal {A}}=\{M_i:i\in N\}$ in M, we aim to show that, if N is $\aleph _0$-categorical and each $M_i$ possess a stronger notion of $\aleph _0$-categoricity, then M is $\aleph _0$-categorical. The stronger notion that we require comes from the following definition, which generalises the notion of $\aleph _0$-categoricity of set extensions.

Definition 3.3

Let M be a structure and $\varPsi$ a subgroup of Aut(M). Then we say that that M is $\aleph _0$-categorical over $\varPsi$ if $\varPsi$ has only finitely many orbits in its action on $M^n$ for each $n\ge 1$. We denote the resulting equivalence relation on $M^n$ as $\sim _{M,\varPsi ,n}$.

By taking $\varPsi$ to be those automorphisms which fix certain subsets of M we recover our original definition of $\aleph _0$-categoricity of a set extension. Similarly, by taking $\varPsi$ to be those automorphisms which preserve a fixed equivalence relation, or those which fix certain equivalence classes, we obtain a pair of notions defined in [9].

Lemma 3.4

Let M be a structure, and ${\mathcal {A}}=\{M_i:i\in N\}$ be an $(M,M';{{\underline{N}}};\varPsi )$-system. If ${{\underline{N}}}$ is $\aleph _0$-categorical and each $M_i$ is $\aleph _0$-categorical over $\varPsi _{i,i}$ then

$$\begin{aligned} |(M')^n/\sim _{M,n}|<\aleph _0 \end{aligned}$$

for each $n\ge 1$.

Proof

Let ${{\underline{N}}}=(N;N_1,\ldots ,N_r)$ and, for each $1\le k \le r$, fix some $m_k\in N_k$. For each $i\in N_k$, let $\theta _i\in \varPsi _{i,m_k}$, noting that such an element exists by Condition (3.1) on $\varPsi$. Let ${{\overline{a}}}=(a_1,\ldots ,a_n)$ and ${{\overline{b}}}=(b_1,\ldots ,b_n)$ be a pair of n-tuples of $M'$, with $a_t\in M_{i_t}$ and $b_t\in M_{j_t}$, and such that $(i_1,\ldots ,i_n) \, \sim _{{{\underline{N}}},n} \, (j_1,\ldots ,j_n)$ via $\pi \in \text {Aut}({{\underline{N}}})$, say. For each $1 \le k\le r$, let $i_{k1},i_{k2},\dots ,i_{kn_k}$ be the entries of $(i_1,\ldots ,i_n)$ belonging to $N_k$, where $k1<k2<\cdots <kn_k$, and set

$$\begin{aligned} {{\overline{a}}}_k=(a_{k1},\ldots ,a_{kn_k})\in (M')^{n_k}. \end{aligned}$$

We similarly form each ${{\overline{b}}}_k$, observing that as $i_t\pi =j_t$ for each $1\le t \le n$ and $\pi$ fixes the sets $N_j$ setwise ($1\le j \le r$) the elements $j_{k1},j_{k2},\dots ,j_{kn_k}$ are precisely the entries of $(j_1,\ldots ,j_n)$ belonging to $N_k$, so that ${{\overline{b}}}_k=(b_{k1},\ldots ,b_{kn_k})$ for some $b_{kt}\in M'$. Notice that as $N_1,\ldots ,N_r$ partition N we have $n=n_1+n_2+\cdots + n_r$. Since $i_{kt},j_{kt}\in N_k$ for each $1 \le t \le n_k$, we have that $a_{kt}\theta _{i_{kt}}$ and $b_{kt}\theta _{j_{kt}}$ are elements of $M_{m_k}$. We may thus suppose further that for each $1\le k \le r$,

$$\begin{aligned} (a_{k1}\theta _{i_{k1}},\ldots ,a_{{kn_k}}\theta _{i_{kn_k}}) \, \sim _{M_{m_k},\varPsi _{m_k,m_k},n_k} \, (b_{k1}\theta _{j_{k1}},\ldots ,b_{{kn_k}}\theta _{j_{kn_k}}) \end{aligned}$$

via $\sigma _k\in \varPsi _{m_k,m_k}$, say (where if ${{\overline{a}}}_k$ is a 0-tuple, then we take $\sigma _k$ to be the identity of $N_{m_k}$). For each $1\le k \le r$ and each $i\in N_k$, let

$$\begin{aligned} \phi _i=\theta _i \sigma _k \theta _{i\pi }^{-1}:M_i\rightarrow M_{i\pi }, \end{aligned}$$

noting that $\phi _i\in \varPsi _{i,i\pi }$ by Conditions (3.2) and (3.3) on $\varPsi$, since $\theta _i,\sigma _k$ and $\theta _{i\pi }$ are elements of $\varPsi$. Hence, by Condition (3.4) on $\varPsi$, there exists an automorphism $\phi$ of M extending each $\phi _i$. For any $1\le k \le r$ and any $1 \le t \le n_k$ we have

$$\begin{aligned} a_{kt}\phi = a_{kt} \phi _{i_{kt}}= a_{kt} \theta _{i_{kt}} \sigma _k \theta _{i_{kt}\pi }^{-1} = b_{kt}\theta _{j_{kt}} \theta _{j_{kt}}^{-1} = b_{kt}, \end{aligned}$$

and so ${{\overline{a}}} \, \sim _{M,n} \, {{\overline{b}}}$ via $\phi$. Since ${{\underline{N}}}$ is $\aleph _0$-categorical and each $M_i$ are $\aleph _0$-categorical over $\varPsi _{i,i}$, the conditions imposed on the tuples ${{\overline{a}}}$ and ${{\overline{b}}}$ have finitely many choices, and so $|(M')^n/\sim _{M,n}|$ is finite. $\square$

By Corollary 2.9, the structure N in the lemma above can simply be a set. In most cases we take $M'=M$, and the result simplifies accordingly by the RNT as follows.

Corollary 3.5

Let M be a structure, and ${\mathcal {A}}=\{M_i:i\in N\}$ be an $(M;{{\underline{N}}};\varPsi )$-system. If ${{\underline{N}}}$ is $\aleph _0$-categorical and each $M_{i}$ is $\aleph _0$-categorical over $\varPsi _{i,i}$, then M is $\aleph _0$-categorical.

Example 3.6

The corollary above could be used to efficiently prove the interplay of $\aleph _0$-categoricity and the greatest 0-direct decomposition of a semigroup with zero [9, Theorem 4.8]. Indeed, if $S=\bigsqcup ^0_{i\in I} S_i$ is the greatest 0-direct decomposition of S, and $I_1,\ldots ,I_n$ is a finite partition of I corresponding to the isomorphism types of the summands of S, then it is a simple exercise to show that ${\mathcal {S}}=\{S_i:i\in I\}$ is an $(S;(I;I_1,\ldots ,I_n);\varPsi )$-system, where $\varPsi$ is the collection of all isomorphisms between summands. Since $(I;I_1,\ldots ,I_n)$ is $\aleph _0$-categorical, it follows by Corollary 3.5 that S is $\aleph _0$-categorical if each $S_i$ is $\aleph _0$-categorical (over $\varPsi _{i,i}=\text {Aut}(S_i)$).

4 Strong semilattices of semigroups

In this section we study the $\aleph _0$-categoricity of strong semilattices of semigroups by making use of our most recent methodology. We are motivated by the work of the author in [22] and [23], where the homogeneity of bands and inverse semigroups are shown to depend heavily on the homogeneity of strong semilattices of rectangular bands and groups, respectively. Recall that a structure is homogeneous if every isomorphism between finitely generated substructures extend to an automorphism. A uniformly locally finite homogeneous structure is $\aleph _0$-categorical [19, Corollary 3.1.3]. Consequently, each homogeneous band is $\aleph _0$-categorical, although the same is not true for homogeneous inverse semigroups.

While there has not yet been a general study into $\aleph _0$-categorical semilattices, a complete classification of countable homogeneous semilattices was completed in [6] and [7]. Since semilattices are uniformly locally finite, this provids us with a countably infinite collection of $\aleph _0$-categorical semilattices. For example, the linear order ${{\mathbb {Q}}}$ is a homogeneous semilattice, and all $\aleph _0$-categorical linear orders are classified in [27].

Let Y be a semilattice. To each $\alpha \in Y$ associate a semigroup $S_{\alpha }$, and assume that $S_{\alpha } \cap S_{\beta } = \emptyset$ if $\alpha \ne \beta$. For each pair $\alpha , \beta \in Y$ with $\alpha \ge \beta$, let $\psi _{\alpha , \beta }:S_{\alpha } \rightarrow S_{\beta }$ be a morphism such that $\psi _{\alpha , \alpha }$ is the identity mapping and if $\alpha \ge \beta \ge \gamma$ then $\psi _{\alpha , \beta } \psi _{\beta , \gamma } = \psi _{\alpha , \gamma }$. On the set $S=\bigcup _{\alpha \in Y} S_{\alpha }$ define a multiplication by

$$\begin{aligned} a * b = (a \psi _{\alpha , \alpha \beta })(b \psi _{\beta , \alpha \beta }) \end{aligned}$$

for $a\in S_{\alpha }, b \in S_{\beta }$, and denote the resulting structure by $S=[Y;S_{\alpha }; \psi _{\alpha , \beta }]$. Then S is a semigroup, and is called a strong semilattice Y of the semigroups $S_{\alpha }$ ($\alpha \in Y$). The semigroups $S_{\alpha }$ are called the components of S. We follow the convention of denoting an element a of $S_{\alpha }$ as $a_{\alpha }$.

The idempotents of $S=[Y;S_{\alpha }; \psi _{\alpha , \beta }]$ are given by $E(S)=\bigcup _{\alpha \in Y} E(S_{\alpha })$, and if E(S) forms a subsemigroup of S then

$$\begin{aligned} E(S) = [Y;E(S_{\alpha });\psi _{\alpha ,\beta }|_{E(S_{\alpha })}]. \end{aligned}$$

We build automorphisms of strong semilattices of semigroups in a natural way using the following well known result. A proof can be found in [21].

Theorem 4.1

Let $S=[Y; S_{\alpha }; \psi _{\alpha , \beta }]$ be a strong semilattices of semigroups. Let $\pi \in \text {Aut}(Y)$ and, for each $\alpha \in Y$, let $\theta _{\alpha }:S_{\alpha }\rightarrow S_{\alpha \pi }$ be an isomorphism. Assume further that for any $\alpha \ge \beta$, the diagram

(4.1)

commutes. Then the map $\theta =\bigcup _{\alpha \in Y} \theta _{\alpha }$ is an automorphism of S, denoted $\theta =[\theta _{\alpha },\pi ]_{\alpha \in Y}$.

We denote the diagram (4.1) by $[\alpha ,\beta ;\alpha \pi ,\beta \pi ]$. The map $\pi$ is called the induced (semilattice) automorphism of Y, denoted $\theta ^Y$.

Unfortunately, not all automorphisms of strong semilattices of semigroups can be constructed as in Theorem 4.1. We shall call a strong semilattice of semigroups S automorphism-pure if every automorphism of S can be constructed as in Theorem 4.1. For example, every strong semilattice of completely simple semigroups is automorphism-pure [20, Lemma IV.1.8], and so both strong semilattices of groups (Clifford semigroups) and strong semilattices of rectangular bands (normal bands) are automorphism-pure.

Let $S=[Y;S_{\alpha };\psi _{\alpha ,\beta }]$ be a strong semilattice of semigroups. We denote the equivalence relation on Y corresponding to isomorphism types of the semigroups $S_{\alpha }$ by $\eta _S$, so that $\alpha \, \eta _S \, \beta \Leftrightarrow S_{\alpha }\cong S_{\beta }.$ We let $Y^S$ denote the set extension of Y given by $Y^S :=(Y;Y/\eta _S)$.

Proposition 4.2

Let $S=[Y;S_{\alpha };\psi _{\alpha ,\beta }]$ be automorphism-pure and $\aleph _0$-categorical. Then each $S_{\alpha }$ is $\aleph _0$-categorical and $Y^S$ is $\aleph _0$-categorical, with $Y/\eta _S$ finite.

Proof

For each $\alpha \in Y$ fix some $x_{\alpha }\in S_{\alpha }$. We claim that $\{(S_{\alpha },x_{\alpha }):\alpha \in Y\}$ forms a system of 1-pivoted p.r.c. subsemigroups of S. Indeed, let $\theta$ be an automorphism of S such that $x_{\alpha }\theta =x_{\beta }$ for some $\alpha ,\beta \in Y$. Since S is automorphism-pure, there exists $\pi \in \text {Aut}(Y)$ and isomorphisms $\theta _{\alpha }:S_\alpha \rightarrow S_{\alpha \pi }$ ($\alpha \in Y$) such that $\theta =[\theta _{\alpha },\pi ]_{\alpha \in Y}$. Hence $S_{\alpha }\theta =S_{\beta }$, and the claim follows. Consequently, by the $\aleph _0$-categoricity of S and Proposition 2.3, each $S_{\alpha }$ is $\aleph _0$-categorical and $Y/\eta _S$ is finite.

Let ${{\overline{a}}}=(\alpha _1,\ldots ,\alpha _n)$ and ${{\overline{b}}}=(\beta _1,\ldots ,\beta _n)$ be a pair of n-tuples of Y such that there exists $a_{\alpha _k}\in S_{\alpha _k}$ and $b_{\beta _k}\in S_{\beta _k}$ with $(a_{\alpha _1}, \dots , a_{\alpha _n}) \, \sim _{S,n} \, (b_{\beta _1},\ldots , b_{\beta _n})$ via $[\theta '_{\alpha },\pi ']_{\alpha \in Y}\in \text {Aut}(S)$, say. Since $\pi '\in \text {Aut}(Y)$ and $S_{\alpha }\cong S_{\alpha \pi '}$ for each $\alpha \in Y$, it follows that $\pi ' \in \text {Aut}(Y^S)$. Moreover, $\alpha _k\pi '=\beta _k$ for each k, so that ${{\overline{a}}} \, \sim _{Y^S,n} \, {{\overline{b}}}$ via $\pi '$. We have thus shown that

$$\begin{aligned} |(Y^S)^n/\sim _{Y^S,n}|\le |S^n/\sim _{S,n}|<\aleph _0, \end{aligned}$$

as S is $\aleph _0$-categorical. Hence $Y^S$ is $\aleph _0$-categorical. $\square$

A natural question arises: how can we build an $\aleph _0$-categorical strong semilattice of semigroups from an $\aleph _0$-categorical semilattice and a collection of $\aleph _0$-categorical semigroups? In this paper we will only be concerned with the $\aleph _0$-categoricity of strong semilattices of semigroups in which all connecting morphisms are injective or all are constant. For arbitrary connecting morphisms, the problem of assessing $\aleph _0$-categoricity appears to be difficult to capture in a reasonable way. Examples of more complex $\aleph _0$-categorical strong semilattices of semigroups arise from Quinn-Gregson [22], where the universal normal band is shown to have surjective but not injective connecting morphisms. We first study the case where each connecting morphism is a constant map.

Suppose that Y is a semilattice and, for each $\alpha \in Y$, $S_{\alpha }$ is a semigroup containing an idempotent $e_{\alpha }$. For each $\alpha \in Y$ let $\psi _{\alpha ,\alpha }$ be the identity automorphism of $S_{\alpha }$, and for $\alpha > \beta$ let $\psi _{\alpha ,\beta }$ be the constant map with image $\{e_{\beta }\}$. We follow the notation of Worawiset [30] and let $\psi _{\alpha ,\beta }:=C_{\alpha ,e_{\beta }}$ for each $\alpha >\beta$ in Y. It is easy to check that $\psi _{\alpha ,\beta }\psi _{\beta ,\gamma }=\psi _{\alpha ,\gamma }$ for all $\alpha \ge \beta \ge \gamma$ in Y, so that $S=[Y;S_{\alpha };C_{\alpha ,e_{\beta }}]$ forms a strong semilattice of semigroups. We call S a constant strong semilattice of semigroups.

Definition 4.3

If $S=[Y;S_{\alpha };C_{\alpha ,e_{\beta }}]$ is a constant strong semilattice of semigroups, then we denote the subset of Iso$(S_{\alpha };S_{\beta })$ consisting of those isomorphisms which map $e_{\alpha }$ to $e_{\beta }$ as Iso$(S_{\alpha };S_{\beta })^{[e_{\alpha };e_{\beta }]}$. Notice that the set Iso$(S_{\alpha };S_{\alpha })^{[e_{\alpha };e_{\alpha }]}$ is simply the subgroup Aut$(S_{\alpha };\{e_{\alpha }\})$ of Aut($S_{\alpha }$). We may then define a relation $\upsilon _S$ on Y by

$$\begin{aligned} \alpha \, \upsilon _S \, \beta \Leftrightarrow \text {Iso}(S_{\alpha };S_{\beta })^{[e_{\alpha };e_{\beta }]}\ne \emptyset , \end{aligned}$$

so that $\upsilon _S\subseteq \eta _S$.

The relation $\upsilon _S$ is reflexive since $1_{S_{\alpha }}\in \text {Aut}(S_{\alpha };\{e_{\alpha }\})$ for each $\alpha \in Y$, and it easily follows that $\upsilon _S$ forms an equivalence relation on Y.

Proposition 4.4

Let $S=[Y;S_{\alpha };C_{\alpha ,e_{\beta }}]$ be such that $Y/\upsilon _S=\{Y_1,\ldots ,Y_r\}$ is finite, ${\mathcal {Y}}=(Y;Y_1,\ldots ,Y_r)$ is $\aleph _0$-categorical and each $S_{\alpha }$ is $\aleph _0$-categorical. Then S is $\aleph _0$-categorical.

Proof

We prove that $\{S_{\alpha }:\alpha \in Y\}$ forms an $(S;{\mathcal {Y}};\varPsi )$-system for some $\varPsi$. For each $\alpha ,\beta \in Y$, let $\varPsi _{\alpha ,\beta }=\text {Iso}(S_{\alpha };S_{\beta })^{[e_{\alpha };e_{\beta }]}$ and fix $\varPsi =\bigcup _{\alpha ,\beta \in Y} \varPsi _{\alpha ,\beta }$. Then Conditions (3.1)–(3.3) are seen to be satisfied since $\upsilon _S$ forms an equivalence relation on Y. Let $\pi \in \text {Aut}({\mathcal {Y}})$ and, for each $\alpha \in Y$, let $\theta _{\alpha }\in \varPsi _{\alpha ,\alpha \pi }$. We claim that $\theta =[\theta _{\alpha },\pi ]_{\alpha \in Y}$ is an automorphism of S. Indeed, for any $s_{\alpha }\in S_{\alpha }$ and any $\beta < \alpha$ we have

$$\begin{aligned} s_{\alpha }C_{\alpha ,e_{\beta }}\theta _{\beta } = e_{\beta }\theta _{\beta }=e_{\beta \pi } = s_{\alpha }\theta _{\alpha } C_{\alpha \pi ,e_{\beta \pi }} \end{aligned}$$

so that the diagram $[\alpha ,\beta ;\alpha \pi ,\beta \pi ]$ commutes. Moreover $[\alpha ,\alpha ;\alpha \pi ,\alpha \pi ]$ commutes as

$$\begin{aligned} s_{\alpha }1_{S_{\alpha }}\theta _{\alpha }=s_{\alpha }\theta _{\alpha } =s_{\alpha }\theta _{\alpha }1_{S_{\alpha \pi }}, \end{aligned}$$

and the claim follows by Theorem 4.1. Since $\theta$ extends each $\theta _{\alpha }$, we have that $\{S_{\alpha }:\alpha \in Y\}$ is an $(S;{\mathcal {Y}};\varPsi )$-system. Moreover, as $S_\alpha$ is $\aleph _0$-categorical, it is $\aleph _0$-categorical over $\varPsi _{\alpha ,\alpha }=\text {Aut}(S_{\alpha };\{e_{\alpha }\})$ by [9, Lemma 2.6]. Hence S is $\aleph _0$-categorical by Corollary 3.5. $\square$

Examining our two main classes of automorphism-pure strong semilattices of semigroups: Clifford semigroups and normal bands, the result above reduces accordingly. If $S=[Y;G_{\alpha };C_{\alpha ,e_{\beta }}]$ is a constant strong semilattice of groups, then $e_{\alpha }$ is the identity of $G_{\alpha }$, and so Iso$(G_{\alpha };G_{\beta })=\text {Iso}(G_{\alpha };G_{\beta })^{[e_{\alpha };e_{\beta }]}$ for each $\alpha ,\beta \in Y$. On the other hand, if $S=[Y;B_{\alpha };C_{\alpha ,e_{\beta }}]$ is a constant strong semilattice of rectangular bands, then it follows from Lemma 2.7 that $\text {Iso}(B_{\alpha };B_{\beta })\ne \emptyset$ if and only if $\text {Iso}(B_{\alpha };B_{\beta })^{[e_{\alpha };e_{\beta }]} \ne \emptyset$, for any $e_{\alpha }\in B_{\alpha },e_{\beta }\in B_{\beta }$. In both cases we therefore have $\upsilon _S=\eta _S$. Moreover, each rectangular band $B_{\alpha }$ is $\aleph _0$-categorical by Theorem 2.8, and the following result is then immediate by Propositions 4.2 and 4.4.

Corollary 4.5

Let $S=[Y;S_{\alpha };C_{\alpha ,e_{\beta }}]$ be a constant strong semilattice of rectangular bands (groups). Then S is $\aleph _0$-categorical if and only if $Y^S$ is $\aleph _0$-categorical, with $Y/\eta _S$ finite (and each group $S_{\alpha }$ is $\aleph _0$-categorical).

We now consider the $\aleph _0$-categoricity of a strong semilattice of semigroups $S=[Y;S_{\alpha };\psi _{\alpha ,\beta }]$ such that each connecting morphism is injective. For each $\alpha >\beta$ in Y, we abuse notation somewhat by denoting the isomorphism $\psi _{\alpha ,\beta }^{-1}|_{\text {Im} \, \psi _{\alpha ,\beta }}$ simply by $\psi _{\alpha ,\beta }^{-1}$. We observe that if $\alpha>\beta >\gamma$ and $x_{\gamma }\in$ Im $\psi _{\alpha ,\gamma }$, say $x_{\gamma }=x_{\alpha }\psi _{\alpha ,\gamma }$, then

$$\begin{aligned} x_{\gamma } \psi _{\alpha ,\gamma }^{-1}\psi _{\alpha ,\beta } = x_{\alpha }\psi _{\alpha ,\gamma } \psi _{\alpha ,\gamma }^{-1}\psi _{\alpha ,\beta } = x_{\alpha }\psi _{\alpha ,\beta }=x_{\gamma }\psi _{\beta ,\gamma }^{-1}. \end{aligned}$$

Hence, on the restricted domain Im $\psi _{\alpha ,\gamma }$, we have

$$\begin{aligned} \psi _{\alpha ,\gamma }^{-1}\psi _{\alpha ,\beta } = \psi _{\beta ,\gamma }^{-1}. \end{aligned}$$

(4.2)

If Y has a zero (i.e. a minimum element under the natural order) we may define an equivalence relation $\xi _S$ on Y by $\alpha \, \xi _S \, \beta$ if and only if $S_{\alpha }\psi _{\alpha ,0}=S_\beta \psi _{\beta ,0}$. If $\alpha \, \xi _S \, \beta$ then $\psi _{\alpha ,0}\psi _{\beta ,0}^{-1}$ is an isomorphism from $S_{\alpha }$ onto $S_{\beta }$, and so $\xi _S\subseteq \eta _S$.

Proposition 4.6

Let $S=[Y;S_{\alpha };\psi _{\alpha ,\beta }]$ be such that each $\psi _{\alpha ,\beta }$ is injective. Let Y be a semilattice with zero and $Y/\xi _S=\{Y_1,\ldots ,Y_r\}$ be finite, with

$$\begin{aligned} \{S_\alpha \psi _{\alpha ,0}:\alpha \in Y\}=\{T_1,\ldots ,T_r\}. \end{aligned}$$

Then S is $\aleph _0$-categorical if both ${\mathcal {Y}}=(Y;Y_1,\ldots ,Y_r)$ and ${\mathcal {S}}_0=(S_0;T_1,\ldots ,T_r)$ are $\aleph _0$-categorical. Moreover, if S is automorphism-pure and $\aleph _0$-categorical, then conversely both ${\mathcal {Y}}$ and ${\mathcal {S}}_0$ are $\aleph _0$-categorical.

Proof

Suppose first that both ${\mathcal {Y}}$ and ${\mathcal {S}}_0$ are $\aleph _0$-categorical. Let ${{\overline{a}}}=(a_{\alpha _1},\ldots ,a_{\alpha _n})$ and ${{\overline{b}}}=(b_{\beta _1},\ldots ,b_{\beta _n})$ be n-tuples of S with $(\alpha _1,\ldots ,\alpha _n) \, \sim _{{\mathcal {Y}},n} \, (\beta _1,\ldots ,\beta _n)$ via $\pi \in \text {Aut}({\mathcal {Y}})$, say. Suppose further that

$$\begin{aligned} (a_{\alpha _1}\psi _{\alpha _1,0},\ldots ,a_{\alpha _n}\psi _{\alpha _n,0}) \, \sim _{{\mathcal {S}}_0,n} \, (b_{\beta _1}\psi _{\beta _1,0},\ldots ,b_{\beta _n}\psi _{\beta _n,0}) \end{aligned}$$

via $\theta _0\in \text {Aut}({\mathcal {S}}_0)$, say. Then for each $\alpha \in Y$ we have $S_\alpha \psi _{\alpha ,0}=S_{\alpha \pi }\psi _{\alpha \pi ,0}$, and so we can take an isomorphism $\theta _{\alpha }:S_{\alpha }\rightarrow S_{\alpha \pi }$ given by

$$\begin{aligned} \theta _\alpha =\psi _{\alpha ,0} \, \theta _0 \, \psi _{\alpha \pi ,0}^{-1}. \end{aligned}$$

For each $\alpha \ge \beta$ in Y, the diagram $[\alpha ,\beta ;\alpha \pi ,\beta \pi ]$ commutes as

$$\begin{aligned} \psi _{\alpha ,\beta } \, {\theta }_{\beta }&= \psi _{\alpha ,\beta } \, ( \psi _{\beta ,0} \, \theta _0 \, \psi _{\beta \pi ,0}^{-1}) = {\psi }_{\alpha ,0} \, {\theta }_0 \, \psi _{\beta \pi ,0}^{-1} \\&= {\psi }_{\alpha ,0} \, {\theta }_0 \, ({\psi }_{\alpha \pi ,0}^{-1} \, {\psi }_{\alpha \pi ,\beta \pi }) = {\theta }_{\alpha } \, \psi _{\alpha \pi ,\beta \pi }, \end{aligned}$$

where the penultimate equality is due to (4.2) as Im $\psi _{\alpha \pi ,0} =$ Im $\psi _{\alpha ,0} =$ (Im $\psi _{\alpha ,0})\theta _0$. Hence $\theta =[\theta _{\alpha },\pi ]_{\alpha \in Y}$ is an automorphism of S by Theorem 4.1. Furthermore,

$$\begin{aligned} a_{\alpha _k}\theta =a_{\alpha _k}\theta _{\alpha _k} = a_{\alpha _k} \psi _{\alpha _k,0} \, \theta _0 \, \psi _{\alpha _k\pi ,0}^{-1} = b_{\beta _k} \psi _{\beta _k,0} \, \psi _{\beta _k,0}^{-1} = b_{\beta _k} \end{aligned}$$

for each $1\le k \le n$, so that ${{\overline{a}}} \, \sim _{S,n} \, {{\overline{b}}}$ via $\theta$. We thus have that

$$\begin{aligned} |S^n/\sim _{S,n}| \le |{\mathcal {Y}}^n/\sim _{{\mathcal {Y}},n}| \cdot |{\mathcal {S}}_0^n/\sim _{{\mathcal {S}}_0,n}|<\aleph _0 \end{aligned}$$

and so S is $\aleph _0$-categorical.

Conversely, suppose S is automorphism-pure and $\aleph _0$-categorical. For each $1\le k \le r$, fix some $\gamma _k\in Y_k$, where we assume without loss of generality that $S_{\gamma _k}\psi _{\gamma _k,0}=T_k$. For each $\alpha \in Y$, fix some $x_{\alpha }\in S_{\alpha }$. Let ${{\overline{a}}}=(\alpha _1,\ldots ,\alpha _n)$ and ${{\overline{b}}}=(\beta _1,\ldots ,\beta _n)$ be n-tuples of Y such that

$$\begin{aligned} (x_{\alpha _1},\ldots ,x_{\alpha _n}, x_{\gamma _1},\ldots ,x_{\gamma _r}) \, \sim _{S,n+r} \, (x_{\beta _1},\ldots ,x_{\beta _n}, x_{\gamma _1},\ldots ,x_{\gamma _r}), \end{aligned}$$

via $\theta \in \text {Aut}(S)$, say. Since S is automorphism-pure there exists $\pi \in \text {Aut}(Y)$ and $\theta _{\alpha }\in \text {Iso}(S_{\alpha };S_{\alpha \pi })$ such that $\theta =[\theta _{\alpha },\pi ]_{\alpha \in Y}$. The automorphism $\pi$ fixes each $\gamma _k$, so that $S_{\gamma _k}\theta =S_{\gamma _k}$. Hence, as the diagram $[\gamma _k,0;\gamma _k,0]$ commutes for each k, we have

$$\begin{aligned} T_k = S_{\gamma _k}\psi _{\gamma _k,0}=(S_{\gamma _k}\theta _{\gamma _k}) \psi _{\gamma _k,0}=S_{\gamma _k}\psi _{\gamma _k,0}\theta _0=T_k\theta _0=T_k\theta . \end{aligned}$$

If $\alpha \in Y_k$ then, by the commutativity of the diagram $[\alpha ;0;\alpha \pi ,0]$, we therefore have

$$\begin{aligned} S_{\alpha }\psi _{\alpha ,0} = T_k= T_k\theta _0=S_{\alpha }\psi _{\alpha ,0}\theta _0 = S_{\alpha }\theta _{\alpha }\psi _{\alpha \pi ,0} = S_{\alpha \pi }\psi _{\alpha \pi ,0}, \end{aligned}$$

and so $\pi \in \text {Aut}({\mathcal {Y}})$. We have shown that

$$\begin{aligned} |{\mathcal {Y}}^n/\sim _{{\mathcal {Y}},n}| \le |S^{n+r}/\sim _{S,n+r}|<\aleph _0 \end{aligned}$$

and so ${\mathcal {Y}}$ is $\aleph _0$-categorical. Now suppose ${{\overline{c}}}$ and ${{\overline{d}}}$ are n-tuples of ${\mathcal {S}}_0$ such that

$$\begin{aligned} ({{\overline{c}}},x_{\gamma _1},\ldots ,x_{\gamma _r}) \, \sim _{S,n+r} \, ({{\overline{d}}},x_{\gamma _1},\ldots ,x_{\gamma _r}), \end{aligned}$$

via $\theta '=[\theta _{\alpha }',\pi ']_{\alpha \in Y}\in \text {Aut}(S)$, say. Then arguing as before we have that $T_k\theta '=T_k$ for each k, and it follows that $\theta _0'\in \text {Aut}({\mathcal {S}}_0)$, with ${{\overline{c}}}\theta _0'={{\overline{d}}}$. Hence

$$\begin{aligned} |{\mathcal {S}}_0^n/\sim _{{\mathcal {S}}_0,n}|\le |S^{n+r}/\sim _{S,n+r}|<\aleph _0 \end{aligned}$$

and so ${\mathcal {S}}_0$ is $\aleph _0$-categorical. $\square$

Note that if Y is finite, then the meet of all the elements of Y is a zero. Moreover, as Y is finite, it is $\aleph _0$-categorical over any set of subsets by the RNT, and so the result above simplifies accordingly in this case:

Corollary 4.7

Let $S=[Y;S_{\alpha };\psi _{\alpha ,\beta }]$ be such that Y is finite and each $\psi _{\alpha ,\beta }$ is injective. If ${\mathcal {S}}_0=({S}_0;\{S_{\alpha }\psi _{\alpha ,0}:\alpha \in Y\})$ is $\aleph _0$-categorical then S is $\aleph _0$-categorical. Conversely, if S is automorphism-pure and $\aleph _0$-categorical then ${\mathcal {S}}_0$ is $\aleph _0$-categorical.

For a Clifford semigroup S, the property that the connecting morphisms are injective is equivalent to S being is E-unitary, that is, such that for all $e\in E(S)$ and all $s\in S$, if $es\in E$ then $s\in E(S)$ [17, Exercise 5.20]. Since Clifford semigroups are automorphism-pure, we therefore have the following simplification of Proposition 4.6.

Corollary 4.8

Let $S=[Y;G_{\alpha };\psi _{\alpha ,\beta }]$ be an E-unitary Clifford semigroup. Let Y be a semilattice with zero and $Y/\xi _S$ be finite. Then S is $\aleph _0$-categorical if and only if $(Y;Y/\xi _S)$ and $(S_0; \{S_\alpha \psi _{\alpha ,0}:\alpha \in Y\})$ are $\aleph _0$-categorical. In particular, if Y is finite then S is $\aleph _0$-categorical if and only if $(S_0; \{S_\alpha \psi _{\alpha ,0}:\alpha \in Y\})$ is $\aleph _0$-categorical.

Example 4.9

We use the work of Apps [1] to construct examples of $\aleph _0$-categorical E-unitary Clifford semigroups as follows. Let G be an $\aleph _0$-categorical group and $H_1<H_2<\cdots$ a characteristic series in G, so that each $H_i$ is a characteristic subgroup of G and $H_i$ is a subgroup of $H_{i+1}$. Apps proved that such a series must be finite, and there exists a characteristic series $\{1\}=G_0<G_1<G_2<\cdots <G_n=G$ with each $G_i/G_{i-1}$ a characteristically simple $\aleph _0$-categorical group. For each $0\le i \le n$, let $K_i=G_i\times \{i\}$ be an isomorphic copy of $G_i$. For each ${0}\le {i} \le j \le {n}$, let $\psi _{i,j}:K_i\rightarrow K_{j}$ be the map given by $(x,i)\psi _{i,j}=(x,j)$. Then we may form a strong semilattice of the groups $K_i$ by taking $S=[Y;K_i;\psi _{i,j}]$, where Y is the set $\{0, 1,\ldots ,n\}$ with the reverse ordering $0>1>2>\cdots >n$. Notice that S is E-unitary as each connecting morphism is injective. Moreover, each $K_i\psi _{i,n}=G_i\times \{n\}$ is a characteristic subgroup of $K_n=G_n\times \{n\}$. Hence, by Lemma 2.6, $(K_n;\{K_i\psi _{i,n}:1\le i \le n \})$ is $\aleph _0$-categorical. Since Y is finite, we have that $(Y;Y/\xi _S)$ is $\aleph _0$-categorical, and so S is $\aleph _0$-categorical by Corollary 4.8.

If $S=[Y;S_{\alpha };\psi _{\alpha ,\beta }]$ is such that each connecting morphism is an isomorphism, then $Y/\xi _S=\{Y\}$, and so the result above simplifies accordingly. However we can prove a more general result directly (without the condition that Y has a zero) with aid of the following proposition. The result is folklore, but a proof can be found in [21].

Proposition 4.10

Let $S=[Y;S_{\alpha };\psi _{\alpha ,\beta }]$ be such that each $\psi _{\alpha ,\beta }$ is an isomorphism. Then $S\cong S_{\alpha } \times Y$ for any $\alpha \in Y$ . Conversely, if T is a semigroup and Z is a semilattice then $T\times Z$ is isomorphic to a strong semilattice of semigroups such that each connecting morphism is an isomorphism.

Corollary 4.11

Let $S=[Y;S_{\alpha };\psi _{\alpha ,\beta }]$ be such that each $\psi _{\alpha ,\beta }$ is an isomorphism. If $S_{\alpha }$ and Y are $\aleph _0$-categorical, then S is $\aleph _0$-categorical. Moreover, if S is automorphism-pure then the converse holds.

Proof

By Proposition 4.10, S is isomorphic to $S_{\alpha }\times Y$ for any $\alpha \in Y$. The first half of the result then follows as $\aleph _0$-categoricity is preserved by finite direct products [12].

If S is automorphism-pure then the converse holds by Proposition 4.2, as $(Y;Y/\eta _S)$ being $\aleph _0$-categorical clearly implies Y is $\aleph _0$-categorical. $\square$

5 $\aleph _0$-categorical Rees matrix semigroups

A semigroup S is called simple (0-simple) if it has no proper ideals (if its only proper ideal is $\{0\}$ and $S^2\ne \{0\}$). A simple (0-simple) semigroup is called completely simple (completely 0-simple) if contains a primitive idempotent, i.e. a non-zero idempotent e such that for any non-zero idempotent f of S,

$$\begin{aligned} ef=fe=f \Rightarrow e=f. \end{aligned}$$

Since an $\aleph _0$-categorical semigroup is periodic, it follows that every $\aleph _0$-categorical (0-)simple semigroup is completely (0)-simple (see the proof of Theorem 3.12 of [9]). By Rees theorem [25], to study the $\aleph _0$-categoricity of a completely 0-simple semigroup, it is sufficient to consider Rees matrix semigroups:

Theorem 5.1

(The Rees Theorem) Let G be a group, let I and $\varLambda$ be non-empty index sets and let $P=(p_{\lambda ,i})$ be an $\varLambda \times I$ matrix with entries in $G\cup \{0\}$. Suppose no row or column of P consists entirely of zeros (that is, P is regular). Let $S=(I\times G \times \varLambda ) \cup \{0\}$, and define multiplication $*$ on S by

$$\begin{aligned}&(i,g,\lambda )*(j,h,\mu ) = \left\{ \begin{array}{ll} (i,g p_{\lambda , j} h,\mu ) &{} \text {if } p_{\lambda , j}\ne 0\\ 0 &{} \text {else } \end{array} \right. \\&0*(i,g,\lambda ) =(i,g,\lambda )*0=0*0=0. \end{aligned}$$

Then S is a completely 0-simple semigroup, denoted ${\mathcal {M}}^0[G;I,\varLambda ;P]$, and is called a (regular) Rees matrix semigroup (over G). Conversely, every completely 0-simple semigroup is isomorphic to a Rees matrix semigroup.

The matrix P is called the sandwich matrix of S. If P has no zero entries, then $I\times G \times \varLambda$ forms a subsemigroup of ${\mathcal {M}}^0[G;I,\varLambda ,P]$, called a Rees matrix semigroup without zero and denoted ${\mathcal {M}}[G;I,\varLambda ;P]$. Every completely simple semigroup is isomorphic to a Rees matrix semigroup without zero [17, Section 3.3].

Lemma 5.2

Let G be a group and P be a $\varLambda \times I$ matrix with entries from G. Then ${\mathcal {M}}[G;I,\varLambda ;P]$ is $\aleph _0$-categorical if and only if ${\mathcal {M}}^0[G;I,\varLambda ;P]$ is $\aleph _0$-categorical.

Proof

The result is immediate from Gould and Quinn-Gregson [9, Corollary 2.12] since ${\mathcal {M}}^0[G;I,\varLambda ;P]$ is isomorphic to ${\mathcal {M}}[G;I,\varLambda ;P]$ with a zero adjoined. $\square$

As a consequence, to examine the $\aleph _0$-categoricity of both completely simple and completely 0-simple semigroups, it suffices to study Rees matrix semigroups.

A fundamental discovery in [9] was that to understand the $\aleph _0$-categoricity of an arbitrary semigroup, it is necessary to study $\aleph _0$-categorical completely (0-)simple semigroups. Indeed, they arise as principal factors of an $\aleph _0$-categorical semigroup, as well as giving examples of 0-direct indecomposable summands in a semigroup with zero.

In [9] the $\aleph _0$-categoricity of Rees matrix semigroups over identity matrices (known as Brandt semigroups) were determined, although we deferred the general case to this current article. Countable homogeneous completely simple semigroups have been classified (modulo our understanding of homogeneous groups) in [24], which gives rise to more complex examples of $\aleph _0$-categorical completely (0-)simple semigroups.

Given a Rees matrix semigroup $S={\mathcal {M}}^0[G;I,\varLambda ;P]$ with $P=(p_{\lambda ,i})$, we let G(P) denote the subset of G of all non-zero entries of P, that is, $G(P): = \{p_{\lambda ,i}: p_{\lambda ,i}\ne 0\}$. The idempotents of S are easily described [17, Page 71]:

$$\begin{aligned} E(S) = \{(i,p_{\lambda ,i}^{-1},\lambda ):p_{\lambda ,i}\ne 0\}. \end{aligned}$$

Since there exists a simple isomorphism theorem for Rees matrix semigroups [17, Theorem 3.4.1] (see Theorem 5.10), we should be hopeful of achieving a thorough understanding of $\aleph _0$-categorical Rees matrix semigroups via the RNT. However, from the isomorphism theorem it is not clear how the $\aleph _0$-categoricity of the semigroup ${\mathcal {M}}^0[G;I,\varLambda ;P]$ affects the sets I and $\varLambda$. We instead follow a technique of Graham [10] and Houghton [15] of constructing a bipartite graph from the sets I and $\varLambda$.

A bipartite graph is a (simple) graph whose vertices can be split into two disjoint non-empty sets L and R such that every edge connects a vertex in L to a vertex in R. The sets L and R are called the left set and the right set, respectively. Formally, a bipartite graph is a triple $\varGamma =\langle L,R, E \rangle$ such that L and R are non-empty trivially intersecting sets and

$$\begin{aligned} E\subseteq \{ \{x,y\} :x\in L, \, y\in R \}. \end{aligned}$$

We call $L\cup R$ the set of vertices of $\varGamma$ and E the set of edges. An isomorphism between a pair of bipartite graphs $\varGamma =\langle L,R,E \rangle$ and $\varGamma '=\langle L',R',E' \rangle$ is a bijection $\psi :L\cup R \rightarrow L'\cup R'$ such that $L\psi =L'$, $R\psi =R'$, and $\{l,r\}\in E$ if and only if $\{l\psi ,r\psi \}\in E'$. We are therefore regarding bipartite graphs in the signature $L_{BG}=\{Q_L,Q_R,E\}$, where $Q_L$ and $Q_R$ are unary relations, which correspond to the sets L and R, respectively, and E is a binary relation corresponding to the edge relation (here we abuse the notation somewhat by letting E denote the edge relation and the set of edges).

Let $\varGamma =\langle L,R, E \rangle$ be a bipartite graph. Then $\varGamma$ is called complete if, for all $x\in L, \, y\in R$, we have $\{x,y\}\in E$. If $E=\emptyset$ then $\varGamma$ is called empty. If each vertex of $\varGamma$ is incident to exactly one edge, then $\varGamma$ is called a perfect matching. The complement of $\varGamma$ is the bipartite graph $\langle L, R,E'\rangle$ with

$$\begin{aligned} E'=\{\{x,y\}: x\in L, y\in R, \{x,y\}\not \in E\}. \end{aligned}$$

Hence an empty bipartite graph is the complement of a complete bipartite graph, and vice-versa. We call $\varGamma$ random if, for each $k,\ell \in {{\mathbb {N}}}$, and for every distinct $x_1,\ldots ,x_k,y_1,\ldots ,y_{\ell }$ in L (in R) there exists infinitely many $u\in R$ ($u\in L$) such that $\{u,x_i\}\in E$ but $\{u,y_j\}\not \in E$ for each $1\le i \le k$ and $1 \le j \le \ell$.

Clearly, for each pair $n,m\in {{\mathbb {N}}}^*={{\mathbb {N}}}\cup \{\aleph _0\}$, there exists a unique (up to isomorphism) complete biparite graph with left set of size n and right set of size m, which we denote as $K_{n,m}$. There also exists a unique, up to isomorphism, perfect matching with left and right sets of size n, denoted $P_n$. Similar uniqueness holds for the empty bipartite graph $E_{n,m}$ with left set of size n and right set of size m, and the complement of the perfect matching $P_n$, which we denote as $CP_n$. Less obviously, any pair of random bipartite graphs are isomorphic [4].

Theorem 5.3

[8] A countable bipartite graph is homogeneous if and only if it is isomorphic to either $K_{n,m},$ $E_{n,m},$ $P_n,$ $CP_n$ for some $n,m\in {{\mathbb {N}}}^*$, or the random bipartite graph.

Since bipartite graphs are relational structures with finitely many relations, homogeneous bipartite graphs are uniformly locally finite,¹ and thus $\aleph _0$-categorical. Unfortunately, no full classification of $\aleph _0$-categorical bipartite graphs exists.

Let $\varGamma =\langle L,R, E \rangle$ be a bipartite graph. A path ${{\mathfrak {p}}}$ in $\varGamma$ is a finite sequence of vertices

$$\begin{aligned} {{\mathfrak {p}}}=(v_0,v_1,\ldots , v_n) \end{aligned}$$

such that $v_i$ and $v_{i+1}$ are adjacent for each $0\le i \le n-1$. For example, if $\{x,y\}$ is an edge in E then both (x, y) and (y, x) are paths in $\varGamma$. A pair of vertices x and y are connected, denoted $x \bowtie y$, if and only if $x=y$ or there exists a path $(v_1,v_2,\ldots , v_n)$ in $\varGamma$ such that $v_1=x$ and $v_n=y$. It is clear that $\bowtie$ is an equivalence relation on the set of vertices of $\varGamma$, and we call the equivalence classes the connected components of $\varGamma$. Each connected component is a sub-bipartite graph of $\varGamma$ under the induced structure, and we let ${\mathcal {C}}(\varGamma )$ denote the set of connected components of $\varGamma$.

Let $\varGamma$ be a bipartite graph with ${\mathcal {C}}(\varGamma ) =\{\varGamma _i:i\in A\}$. For any automorphism $\phi$ of $\varGamma$ and $x,y\in \varGamma$ we have that $(x,v_2,\ldots ,v_{n-1},y)$ is a path in $\varGamma$ if and only if $(x\phi ,v_2\phi ,\ldots ,v_{n-1}\phi ,y\phi )$ is a path in $\varGamma$, since $\phi$ preserves edges and non-edges. Hence $x \, \bowtie \, y$ if and only if $x\phi \, \bowtie \, y\phi$, and so there exists a bijection $\pi$ of A such that $\varGamma _i\phi =\varGamma _{i\pi }$ for each $i\in I$. We have thus proven the reverse direction of the following result, the forward being immediate.

Proposition 5.4

Let $\varGamma =\langle L,R, E \rangle$ be a bipartite graph with ${\mathcal {C}}(\varGamma )=\{\varGamma _i:i\in A\}$ . Let $\pi$ be a bijection of A and $\phi _i:\varGamma _i\rightarrow \varGamma _{i\pi }$ an isomorphism for each $i\in A$ . Then $\bigcup _{i\in I} \phi _i$ is an automorphism of $\varGamma$ . Conversely, every automorphism of $\varGamma$ can be constructed in this way.

Proposition 5.5

Let $\varGamma =\langle L,R, E \rangle$ be a bipartite graph with ${\mathcal {C}}(\varGamma )=\{\varGamma _i:i\in A\}$. Then $\varGamma$ is $\aleph _0$-categorical if and only if each connected component is $\aleph _0$-categorical and ${\mathcal {C}}(\varGamma )$ is finite, up to isomorphism.

Proof

($\Rightarrow$) By Proposition 5.4 we have that, for any choice of $x_i\in \varGamma _i$ ($i\in A$), the set $\{(\varGamma _i,x_i):i\in A\}$ forms a system of 1-pivoted p.r.c. sub-bipartite graphs of $\varGamma$. The result then follows from Proposition 2.3.

($\Leftarrow$) First we show that ${\mathcal {C}}(\varGamma )$ forms a $(\varGamma ;{{\underline{A}}};\varPsi )$-system in $\varGamma$ for some ${{\underline{A}}}$ and $\varPsi$. Let $A_1,\ldots ,A_r$ be the finite partition of A corresponding to the isomorphism types of the connected components of $\varGamma$, that is, $\varGamma _i\cong \varGamma _j$ if and only if $i,j\in A_k$ for some k. Fix ${{\underline{A}}}=(A;A_1,\ldots ,A_r)$. For each $i,j\in A$, let $\varPsi _{i,j}=\text {Iso}(\varGamma _i;\varGamma _j)$ and fix $\varPsi =\bigcup _{i,j\in A} \varPsi _{i,j}$. Then $\varPsi$ clearly satisfy Conditions (3.1)–(3.3). Let $\pi \in \text {Aut}({{\underline{A}}})$ and, for each $i\in A$, let $\phi _i\in \varPsi _{i,i\pi }$. Then by Proposition 5.4, $\phi =\bigcup _{i\in A} \phi _i$ is an automorphism of $\varGamma$, and so $\varPsi$ satisfies Condition (3.4). Hence ${\mathcal {C}}(\varGamma )$ forms an $(\varGamma ;{{\underline{A}}};\varPsi )$-system. Each $\varGamma _i$ is $\aleph _0$-categorical (over $\varPsi _{i,i}=\text {Aut}(\varGamma _i)$) and ${{\underline{A}}}$ is $\aleph _0$-categorical by Corollary 2.9, and so $\varGamma$ is $\aleph _0$-categorical by Corollary 3.5. $\square$

Definition 5.6

Let $S={\mathcal {M}}^0[G;I,\varLambda ; P]$ be a Rees matrix semigroup with $P=(p_{\lambda ,i})$. Then we form a bipartite graph $\varGamma (P)=\langle I, \varLambda , E\rangle$ with edge set

$$\begin{aligned} E=\{\{i,\lambda \}: p_{\lambda ,i} \ne 0\}, \end{aligned}$$

which we call the induced bipartite graph of S.

The above construct has long been fundamental to the study of Rees matrix semigroups, and has its roots in a paper by Graham in [10]. Here, it is used to describe the maximal nilpotent subsemigroups of a Rees matrix semigroup, where a semigroup is nilpotent if some power is equal to $\{0\}$. All maximal subsemigroups of a finite Rees matrix semigroup were described in the same paper, a result which was later extended in [11] to arbitrary finite semigroups. In [16], Howie used the induced bipartite graph to describe the subsemigroup of a Rees matrix semigroup generated by its idempotents. Finally, in [15], Houghton described the homology of the induced bipartite graph, and a detailed overview of his work is given in [26].

Example 5.7

Let $S={\mathcal {M}}^0[G;\{1,2,3\},\{\lambda ,\mu \}; P]$ where

Then the induced bipartite graph of S is given in Fig. 1.

Fig. 1

Induced bipartite graph

Example 5.8

Let $S={\mathcal {M}}^0[G;I,\varLambda ;P]$ be such that P has no zero entries, so that S is isomorphic to a completely simple semigroup with zero adjoined. Then $\varGamma (P)$ is a complete bipartite graph.

Notation 5.9

Let $S={\mathcal {M}}^0[G;I,\varLambda ;P]$ be a Rees matrix semigroup. For an n-tuple ${{\overline{a}}}=((i_1,g_1,\lambda _1),\ldots ,(i_n,g_n,\lambda _n))$ of $S^*$, we write $\varGamma ({{\overline{a}}})$ for the 2n-tuple $(i_1,\lambda _1,\ldots ,i_n,\lambda _n)$ of $\varGamma (P)$.

Following [2], we adapt the isomorphism theorem for Rees matrix semigroups to explicitly highlight the role of the induced bipartite graph:

Theorem 5.10

Let $S_1={\mathcal {M}}^0[G_1; I_1, \varLambda _1;P_1]$ and $S_2={\mathcal {M}}^0[G_2;I_2,\varLambda _2;P_2]$ be a pair of Rees matrix semigroups with sandwich matrices $P_1=(p_{\lambda , i})$ and $P_2=(q_{\mu , j})$ , respectively. Let $\psi \in \text {Iso}(\varGamma (P_1);\varGamma (P_2))$ , $\theta \in \text {Iso}(G_1;G_2)$ , and $u_i, v_{\lambda }\in G_2$ for each $i\in I_1, \lambda \in \varLambda _1$ . Then the mapping $\phi :S_1\rightarrow S_2$ given by

$$\begin{aligned} (i,g,\lambda )\phi = (i\psi , u_i (g\theta ) v_{\lambda }, \lambda \psi ) \end{aligned}$$

is an isomorphism if and only if $p_{\lambda , i} \, \theta = v_{\lambda } \cdot q_{\lambda \psi , i\psi } \cdot u_i$ whenever $p_{\lambda , i}\ne 0$ . Moreover, every isomorphism from $S_1$ to $S_2$ can be described in this way.

The isomorphism $\phi$ will be denoted as $(\theta ,\psi ,(u_i)_{i\in I},(v_{\lambda })_{\lambda \in \varLambda })$. We also denote the induced group isomorphism $\theta$ as $\phi _{G_1}$, and the induced bipartite graph isomorphism $\psi$ as $\phi _{\varGamma (P_1)}$, so that $\phi =(\phi _{G_1},\psi _{\varGamma (P_1)},(u_i)_{i\in I_1},(v_{\lambda })_{\lambda \in \varLambda _1})$. Note that the induced group isomorphism is not uniquely defined by $\phi$. That is, there may exist $\theta '\in \text {Iso}(G_1;G_2)$ and $u_i',v_{\lambda }'\in G_2$, such that $\theta '\ne \theta$ but $\phi =(\theta ',\psi ,(u_i')_{i\in I_1},(v_{\lambda }')_{\lambda \in \varLambda _1})$. Examples of this phenomenon will occur throughout this work.

The composition and inverses of isomorphisms between Rees matrix semigroups behave in a natural way as follows, and a proof can be found in [21].

Corollary 5.11

Let $S_k={{\mathscr {M}}}^0[G_k;I_k,\varLambda _k;P_k]$ ($k=1,2,3$) be Rees matrix semigroups. Then for any pair of isomorphisms $\phi =(\theta ,\psi ,(u_i)_{i\in I_1},(v_{\lambda })_{\lambda \in \varLambda _1})\in \text {Iso}(S_1;S_2)$ and $\phi '=(\theta ',\psi ',(u_j')_{j\in I_2},(v_{\mu }')_{\mu \in \varLambda _2})\in \text {Iso}(S_2;S_3)$ we have:

• (i) $\phi \phi '=\big (\theta \theta ',\psi \psi ',(u'_{i\psi }(u_i\theta '))_{i\in I_1},((v_{\lambda }\theta ')v_{\lambda \psi }')_{\lambda \in \varLambda _1} \big )$;

• (ii) $\phi ^{-1}=(\theta ^{-1},\psi ^{-1},((u_{i\psi ^{-1}})^{-1}\theta ^{-1})_{i\in I_2},((v_{\lambda \psi ^{-1}})^{-1}\theta ^{-1})_{\lambda \in \varLambda _2})$.

Let $\varGamma =\langle L,R,E \rangle$ be a bipartite graph. For each $n\in {{\mathbb {N}}}$, we let $\sigma _{\varGamma ,n}$ be the equivalence relation on $\varGamma ^n$ given by

$$\begin{aligned} (x_1,\ldots ,x_n) \, \sigma _{\varGamma ,n} \, (y_1,\ldots ,y_n) \Leftrightarrow [x_i\in L \Leftrightarrow y_i\in L, \text { for each } 1\le i \le n]. \end{aligned}$$

Since each entry of an n-tuple of $\varGamma$ lies in either L or R we have that

$$\begin{aligned} |\varGamma ^n/\sigma _{\varGamma ,n}|=2^n, \end{aligned}$$

for each n. Moreover, as the automorphisms of $\varGamma$ fixes the sets L and R, it easily follows that $\sim _{\varGamma ,n} \, \subseteq \, \sigma _{\varGamma ,n}$.

Proposition 5.12

If $S= {\mathcal {M}}^0[G; I, \varLambda ;P]$ is $\aleph _0$-categorical, then G and $\varGamma (P)$ are $\aleph _0$-categorical.

Proof

Since G is isomorphic to the non-zero maximal subgroups of S, it is $\aleph _0$-categorical by Gould and Quinn-Gregson [9, Corollary 3.7]. Now let ${{\overline{a}}}=(a_1,\ldots ,a_n)$ and ${{\overline{b}}}=(b_1,\ldots ,b_n)$ be a pair of $\sigma _{\varGamma (P),n}$-related n-tuples of $\varGamma (P)$. Let $i_1<i_2<\cdots <i_s$ and $j_1<j_2<\cdots <j_t$ be the indexes of entries of ${{\overline{a}}}$ lying in I and $\varLambda$, respectively (noting that the same is true for ${{\overline{b}}}$ as ${{\overline{a}}} \, \sigma _{\varGamma ,n} \, {{\overline{b}}}$). Suppose further that there exists $i\in I,\lambda \in \varLambda$ such that the n-tuples

$$\begin{aligned}&((a_{i_1},1,\lambda ), \dots , (a_{i_s},1,\lambda ),(i,1,a_{j_1}),\ldots ,(i,1,a_{j_t})) \quad \text {and} \\&((b_{i_1},1,\lambda ), \dots , (b_{i_s},1,\lambda ),(i,1,b_{j_1}),\ldots ,(i,1,b_{j_t})), \end{aligned}$$

are automorphically equivalent via $\phi \in \text {Aut}(S)$, say. By Theorem 5.10, $a_{i_r}\phi _{\varGamma (P)}=b_{i_r}$ and $a_{j_{r'}}\phi _{\varGamma (P)}=b_{j_{r'}}$ for each $1\le r \le s$ and $1\le r'\le t$. Hence ${{\overline{a}}} \, \sim _{\varGamma (P),n} \, {{\overline{b}}}$ via $\phi _{\varGamma (P)}$, and we have thus shown that

$$\begin{aligned} |\varGamma (P)^{n}/\sim _{\varGamma (P),n}| \le 2^{n} \cdot |S^n/\sim _{S,n}|. \end{aligned}$$

Hence $\varGamma (P)$ is $\aleph _0$-categorical by the $\aleph _0$-categoricity of S. $\square$

However, the converse to the proposition above does not hold in general (even in the completely simple case).

Example 5.13

Let $G=\{1,a\}$ be the group of size 2 and let $I=\{i_0,i_1,\ldots ,\}$ and $\varLambda =\{\lambda _0,\lambda _1,\ldots ,\}$ be infinite sets. Let P be the $\varLambda \times I$ matrix in which $p_{\lambda _k,i_{\ell }}=a$ if and only if $k\ge \ell \ge 1$, that is,

$$\begin{aligned} P = \begin{bmatrix} 1 &{} 1 &{} 1 &{} 1 &{} \cdots &{} 1 &{} \cdots \\ 1&{} a &{} 1 &{} 1 &{} \cdots &{} 1 &{} \cdots \\ 1 &{} a &{} a &{} 1 &{} \ddots &{} \ddots &{} \vdots &{} \\ \vdots &{} \vdots &{} \ddots &{} \ddots &{} 1 &{} 1 &{} \cdots \\ 1 &{} a &{} \cdots &{} a &{} a &{} 1 &{} \cdots \\ 1 &{} a&{} \cdots &{} a &{} a &{} a &{} \cdots \\ \vdots &{} \vdots &{} \cdots &{} \vdots &{} \vdots &{} \vdots &{} \ddots \end{bmatrix}. \end{aligned}$$

Let $S={\mathcal {M}}[G;I,\varLambda ;P]$. Then $\varGamma (P)$ is a complete bipartite graph, and thus $\aleph _0$-categorical. However, $\{((i_0,1,\lambda _0),(i_k,1,\lambda _k)):k\in {{\mathbb {N}}}\}$ can be shown to be an infinite set of distinct 2-automorphism types of S. Alternatively, we will show at the end of the section that S is not $\aleph _0$-categorical by Proposition 5.29.

5.1 Connected Rees components

Let $S_k={\mathcal {M}}^0[G;I_k,\varLambda _k;P_k]$ ($k\in A$) be a collection of Rees matrix semigroups with $P_k=(p_{\lambda ,i}^{(k)})$ and $S_k\cap S_{\ell }=\{0\}$ for each $k,\ell \in A$. Then we may form a single Rees matrix semigroup $S={\mathcal {M}}^0[G;I,\varLambda ;P]$, where $I=\bigcup _{k\in A} I_k$, $\varLambda =\bigcup _{k\in A} \varLambda _k$ and $P=(p_{\lambda ,i})$ is the $\varLambda$ by I matrix defined by

$$\begin{aligned} p_{\lambda ,i} = \left\{ \begin{array}{ll} p_{\lambda ,i}^{(k)} &{} \text {if } \lambda ,i\in \varGamma (P_k), \text { for some } k\\ 0 &{} \text {else. } \end{array} \right. \end{aligned}$$

That is, P is the block matrix

$$\begin{aligned} P = \begin{bmatrix} P_1 &{} 0 &{} 0 &{} \cdots \\ 0&{}P_2 &{} 0 &{} \cdots \\ 0 &{} 0 &{} P_3 &{} \ddots \\ \vdots &{} \vdots &{} \ddots &{} \ddots \end{bmatrix}. \end{aligned}$$

(5.1)

We denote S by ${{\circledast }}_{k\in A}^G S_k$. The subsemigroups $S_k$ of S are called Rees components of S. Notice that each $\varGamma (P_k)$ is a union of connected components of $\varGamma (P)$. The subsemigroup $S_k$ will be called a connected Rees component of S if $\varGamma (P_k)$ is connected (and is therefore a connected component of $\varGamma (P)$).

Conversely, for any Rees matrix semigroup $S={\mathcal {M}}^0[G;I,\varLambda ;P]$ there exists partitions $\{I_k:k\in A\}$ and $\{\varLambda _k:k\in A\}$ of I and $\varLambda$, respectively, such that ${\mathcal {C}}(\varGamma (P))=\{\varLambda _k\cup I_k:k\in A\}$. Consequently, for each $k\in A$, the subsemigroup $S_k={\mathcal {M}}^0[G;I_k,\varLambda _k;P_k]$ of S is a connected Rees component, where $P_k$ is the $\varLambda _k\times I_k$ submatrix of P, and are such that $S_kS_{\ell }=0$ for all $k\ne \ell$. Following the work of Graham [10], we may then permute the rows and columns of P if necessary to assume without loss of generality that P is a block matrix of the form (5.1).

Note that if S is a Rees matrix semigroup with connected Rees components $\{S_k:k\in A\}$ then clearly

$$\begin{aligned} E(S) = \bigcup _{k\in A} E(S_k). \end{aligned}$$

(5.2)

Using the fact that automorphisms of $\varGamma (P)$ arise as collections of isomorphisms between its connected components, we obtain an alternative description of automorphisms of a Rees matrix semigroups. The proof is a simple exercise, and can be found in [21].

Corollary 5.14

Let $S={{\circledast }}^G_{k\in A} S_k ={\mathcal {M}}^0[G;I,\varLambda ;P]$ be a Rees matrix semigroup such that each $S_k={\mathcal {M}}^0[G;I_k,\varLambda _k;P_k]$ is a connected Rees component of S . Let $\pi$ be a bijection of A and, for each $k\in A$ , let $\phi _k=(\theta ,\psi _k,(u_i^{(k)})_{i\in I_k},(v_{\lambda }^{(k)})_{\lambda \in \varLambda _k})$ be an isomorphism from $S_k$ to $S_{k\pi }$ . Then $\phi =(\theta ,\psi ,(u_i)_{i\in I},(v_{\lambda })_{\lambda \in \varLambda })$ is an automorphism of S , where $\psi =\bigcup _{k\in A} \psi _k$ , and if $i,\lambda \in \varGamma (P_k)$ then $u_i=u_i^{(k)}$ and $v_{\lambda }=v_{\lambda }^{(k)}$ . Moreover, every automorphism of S can be described in this way.

We observe that the induced group automorphisms of the isomorphisms $\phi _k$ above must all be equal.

Recall that if $S={\mathcal {M}}^0[G;I,\varLambda ;P]$ is $\aleph _0$-categorical, then $\varGamma (P)$ is $\aleph _0$-categorical by Proposition 5.12, and thus ${\mathcal {C}}(\varGamma (P))$ is finite, up to isomorphism, with each connected component being $\aleph _0$-categorical by Proposition 5.5. We extend this result to the set of all connected Rees components of S as follows:

Proposition 5.15

Let $S={{\circledast }}_{k\in A}^G S_k$ be an $\aleph _0$-categorical Rees matrix semigroup such that each $S_k$ is a connected Rees component of S. Then each $S_k$ is $\aleph _0$-categorical and S has finitely many connected Rees components, up to isomorphism.

Proof

We claim that $\{(S_k,a_k):k\in A\}$ is a system of 1-pivoted p.r.c. subsemigroups of S for any $a_k\in S_k^*$, to which the result follows by Proposition 2.3. Indeed, let $\phi$ be an automorphism of S such that $a_k \phi =a_l$ for some k, l. Then, by Corollary 5.14, there exists a bijection $\pi$ of A with $S_k\phi =S_{k\pi }=S_l$ as required. $\square$

Our interest is now in attaining a converse to the proposition above, since it would provide us with a method for building ‘new’ $\aleph _0$-categorical Rees matrix semigroups from ‘old’. With the aid of Lemma 3.4, we shall prove that a converse exists in the class of Rees matrix semigroups over finite groups. The case where the maximal subgroups are infinite is an open problem.

Given a pair $S={\mathcal {M}}^0[G;I,\varLambda ;P]$ and $S'={\mathcal {M}}^0[G;I',\varLambda ';Q]$ of Rees matrix semigroups over a group G, we denote $\text {Iso}(S;S')(1_{G})$ as the set of isomorphisms between S and $S'$ with trivial induced group isomorphism. That is, $\text {Iso}(S;S')(1_{G})$ is the subset of $\text {Iso}(S;S')$ given by

$$\begin{aligned} \{\phi : \exists \psi \in \text {Iso}(\varGamma (P);\varGamma (Q)) \text { and } u_i, v_{\lambda }\in G \text { such that } \phi = (1_G,\psi ,(u_i)_{i\in I},(v_{\lambda })_{\lambda \in \varLambda })\}. \end{aligned}$$

If $S=S'$ we denote this simply as Aut$(S)(1_G)$, and notice that Aut$(S)(1_G)$ is a subgroup of Aut(S) by Corollary 5.11.

Lemma 5.16

Let $S={\mathcal {M}}^0[G;I,\varLambda ;P]$ be a Rees matrix semigroup over a finite group G. Then S is $\aleph _0$-categorical if and only if S is $\aleph _0$-categorical over Aut$(S)(1_G)$.

Proof

Let S be $\aleph _0$-categorical with $G=\{g_1,\ldots ,g_r\}$ finite. Let ${{\overline{a}}}$ and ${{\overline{b}}}$ be a pair of n-tuples of S. For some fixed $p_{\mu ,j}\ne 0$, let ${{\overline{g}}}$ be the r-tuple of S given by ${{\overline{g}}}=((j,g_1,\mu ),\ldots ,(j,g_r,\mu ))$, and suppose that $({{\overline{a}}},{{\overline{g}}})\, \sim _{S,n+r} \, ({{\overline{b}}},{{\overline{g}}})$ via $\phi =(\theta ,\psi ,(u_i)_{i\in I},(v_{\lambda })_{\lambda \in \varLambda })$, say. Then, for each $1\le k\le r$, we have

$$\begin{aligned} (j,g_k,\mu )\phi =(j\psi ,u_j(g_k\theta )v_{\mu },\mu \psi )=(j,g_k,\mu ), \end{aligned}$$

so that $g_k\theta =u_j^{-1}g_k v_{\mu }^{-1}$. For each $i\in I, \lambda \in \varLambda$, let ${{\bar{u}}}_i=u_iu_j^{-1}$ and ${{\bar{v}}}_{\lambda }=v_{\mu }^{-1}v_{\lambda }$. Then

$$\begin{aligned} (i\psi , {{\bar{u}}}_i g_k {{\bar{v}}}_\lambda ,\lambda \psi )&=(i\psi , (u_iu_j^{-1}) g_k (v_{\mu }^{-1}v_\lambda ),\lambda \psi ) \\&=(i\psi ,u_i(g_k\theta )v_{\lambda },\lambda \psi ) \\&=(i,g_k,\lambda )\phi , \end{aligned}$$

for any $(i,g_k,\lambda )\in S$, so that $\phi =(1_G,\psi ,({{\bar{u}}}_i)_{i\in I},({{\bar{v}}}_\lambda )_{\lambda \in \varLambda })\in \text {Aut}(S)(1_G)$. Consequently, $({{\overline{a}}},{{\overline{g}}})\, \sim _{S,\text {Aut}(S)(1_G),n+r} \, ({{\overline{b}}},{{\overline{g}}})$ and in particular ${{\overline{a}}}\, \sim _{S,\text {Aut}(S)(1_G),n} \, {{\overline{b}}}$. We have thus shown that

$$\begin{aligned} |S^n/\sim _{S,\text {Aut}(S)(1_G),n}|\le |S^{n+r}/\sim _{S,n+r}|<\aleph _0, \end{aligned}$$

as S is $\aleph _0$-categorical. Hence S is $\aleph _0$-categorical over Aut$(S)(1_G)$.

The converse is immediate. $\square$

We are now able to prove our desired converse to Proposition 5.15 in the case where the maximal subgroups are finite.

Theorem 5.17

Let $S={\mathcal {M}}^0[G;I,\varLambda ;P]$ be a Rees matrix semigroup such that G is finite. Then S is $\aleph _0$-categorical if and only if each connected Rees component of S is $\aleph _0$-categorical and S has only finitely many connected Rees components, up to isomorphism.

Proof

($\Rightarrow$) Immediate from Proposition 5.15.

($\Leftarrow$) Since S is regular with finite maximal subgroups, to prove S is $\aleph _0$-categorical, it suffices by [9, Corollary 3.14] to show that $|E(S)^n/\sim _{S,n}|$ is finite, for each $n\in {{\mathbb {N}}}$. Let $\{S_k:k\in A\}$ be the set of connected Rees components of S, which is finite up to isomorphism and with each $S_k$ being $\aleph _0$-categorical. Define a relation $\eta$ on A by $i \, \eta \, j$ if and only if $\text {Iso}(S_i;S_j)(1_G)\ne \emptyset$. By Corollary 5.11 we have that $\eta$ is an equivalence relation.

We first prove that $A/\eta$ is finite. Suppose for contradiction that there exists an infinite set X of pairwise $\eta$-inequivalent elements of A. Since S has finitely many connected components up to isomorphism, there exists an infinite subset $\{i_r:r\in {{\mathbb {N}}}\}$ of X such that $S_{i_n}\cong S_{i_m}$ for each n, m. Fix an isomorphism $\phi _{i_n}:S_{i_n}\rightarrow S_{i_1}$ for each $n\in {{\mathbb {N}}}$. Then as Aut(G) is finite there exists distinct n, m such that $\phi _{i_n}^G=\phi _{i_m}^G$, and so $\phi _{i_n}\phi _{i_m}^{-1}\in \text {Iso}(S_{i_n};S_{j_m})(1_G)$ by Corollary 5.11. Hence $i_n \, \eta \, i_m$, a contradiction, and so $A/\eta$ is finite.

Let $S'=\bigcup _{k\in A} S_k$, noting that $S'$ is the 0-direct union of the $S_k$, and in particular is a subsemigroup of S. Let $A/\eta = \{A_1,\ldots ,A_r\}$ and set ${{\underline{A}}}=(A;A_1,\ldots ,A_r)$. For each $i,j\in A$, let $\varPsi _{i,j}= \text {Iso}(S_i;S_j)(1_G)$ and fix $\varPsi =\bigcup _{i,j\in {A}}\varPsi _{i,j}$. We prove that $\{S_k:k\in A\}$ forms an $(S;S'; {{\underline{A}}};\varPsi )$-system in S. First, by our construction, if $i,j\in A_m$ for some m then $\varPsi _{i,j} \ne \emptyset$, and so $\varPsi$ satisfies Condition (3.1). Furthermore, it follows immediately from Corollary 5.11 that $\varPsi$ satisfies Conditions (3.2) and (3.3). Finally, take any $\pi \in \text {Aut}({{\underline{A}}})$ and, for each $k\in A$, let $\phi _k\in \varPsi _{k,k\pi }$. Then as $\phi _k^G=1_G$ for each $k\in A$, we may construct an automorphism $\phi$ of S from the set of isomorphisms $\{\phi _k:k\in A\}$ by Corollary 5.14. Hence, as $\phi$ extends each $\phi _k$ by construction, we have that $\{S_k:k\in A\}$ forms an $(S;S';{{\underline{A}}};\varPsi )$-system as required. Since $S_k$ is $\aleph _0$-categorical, it is $\aleph _0$-categorical over $\varPsi _{k,k}=\text {Aut}(S_k)(1_G)$ by Lemma 5.16. By Corollary 2.9${{\underline{A}}}$ is $\aleph _0$-categorical, and so

$$\begin{aligned} |(S')^n/\sim _{S,n}|<\aleph _0 \end{aligned}$$

by Lemma 3.4. Given that $E(S) \subseteq S'$ by (5.2), we therefore have that

$$\begin{aligned} |E(S)^n/\sim _{S,n}|\le |(S')^n/\sim _{S,n}|<\aleph _0. \end{aligned}$$

Hence S is $\aleph _0$-categorical. $\square$

Open Problem 5.18

Does Theorem 5.17 hold if G is allowed to be any $\aleph _0$-categorical group?

5.2 Labelled bipartite graphs

In Example 5.13, the problem which arose was that by shifting from the sandwich matrix $P=(p_{\lambda ,i})$ to the induced bipartite graph $\varGamma (P)$ we have “forgotten” the value of the entries $p_{\lambda ,i}$. In this subsection we extend the construction of the induced bipartite graph of a Rees matrix semigroup to attempt to rectifying this problem, as well as to build classes of $\aleph _0$-categorical Rees matrix semigroups. Further examples of $\aleph _0$-categorical Rees matrix semigroups can then be built using Theorem 5.17.

Definition 5.19

Let $\varGamma =\langle L,R,E \rangle$ be a bipartite graph, $\varSigma$ a set, and $f:E\rightarrow \varSigma$ a surjective map. Then the triple $(\varGamma ,\varSigma ,f)$ is called a $\varSigma$-labeled (by f) bipartite graph, which we denote as $\varGamma ^f$.

A pair of $\varSigma$-labeled bipartite graphs $\varGamma ^f=(\varGamma , \varSigma ,f)$ and $\varGamma ^{f'}=(\varGamma ', \varSigma ,f')$ are isomorphic if there exists an isomorphism $\psi :\varGamma \rightarrow \varGamma '$ which preserves labels, that is, such that

$$\begin{aligned} \{x,y\} f=\sigma \Leftrightarrow \{x\psi ,y\psi \}f'=\sigma . \end{aligned}$$

This gives rise to a natural signature in which to consider $\varSigma$-labeled bipartite graphs as follows. For each $\sigma \in \varSigma$, take a binary relation symbol $E_\sigma$ and let

$$\begin{aligned} L_{BG\varSigma }=L_{BG} \cup \{E_{\sigma }:\sigma \in \varSigma \}. \end{aligned}$$

Then we call $L_{BG\varSigma }$ the signature of $\varSigma$-labeled bipartite graphs, where $(x,y)\in E_\sigma$ if and only if $\{x,y\}\in E$ and $\{x,y\}f=\sigma$.

Let $\varGamma ^f$ be a $\varSigma$-labeled bipartite graph. Then for any set $\varSigma '$ and bijection $g:\varSigma \rightarrow \varSigma '$, we can form a $\varSigma '$-labeling of $\varGamma$ simply by taking $\varGamma ^{fg}$, which we call a relabeling of $\varGamma ^f$. Notice that if $\psi$ is an automorphism of $\varGamma$, then $\psi \in \text {Aut}(\varGamma ^f)$ if and only if $\psi \in \text {Aut}(\varGamma ^{fg})$. Indeed, if $\psi \in \text {Aut}(\varGamma ^f)$ then for any edge $\{x,y\}$ of $\varGamma$ we have

$$\begin{aligned} \{x,y\}fg=\sigma ' \Leftrightarrow \{x,y\}f=\sigma 'g^{-1} \Leftrightarrow \{x\psi ,y\psi \}f=\sigma 'g^{-1} \Leftrightarrow \{x\psi ,y\psi \}fg = \sigma ', \end{aligned}$$

since g is a bijection. The converse is proven similarly, and the following result is then immediate from the RNT.

Lemma 5.20

Let $\varGamma ^f$ be a $\varSigma$-labeling of a bipartite graph $\varGamma$. Then $\varGamma ^f$ is $\aleph _0$-categorical if and only if any relabeling of $\varGamma ^f$ is $\aleph _0$-categorical.

Lemma 5.21

If $\varGamma ^f=(\varGamma , \varSigma ,f)$ is an $\aleph _0$-categorical labeled bipartite graph then $\varSigma$ is finite and $\varGamma$ is $\aleph _0$-categorical.

Proof

For each $\sigma \in \varSigma$, let $\{x_{\sigma },y_{\sigma }\}$ be an edge in $\varGamma$ such that $\{x_{\sigma },y_\sigma \}f=\sigma$. Then $\{(x_{\sigma },y_\sigma ):\sigma \in \varSigma \}$ is a set of distinct 2-automorphism types of $\varGamma ^f$, and so $\varSigma$ is finite by the RNT. Since automorphisms of $\varGamma ^f$ induce automorphisms of $\varGamma$, the final result is immediate from the RNT. $\square$

A consequence of the previous pair of lemmas is that, in the context of $\aleph _0$-categoricity, it suffices to consider finitely labeled bipartite graphs, with labeling set $\mathbf{m}=\{1,2,\ldots ,m\}$ for some $m\in {{\mathbb {N}}}$.

Lemma 5.22

Let $\varGamma ^f=(\langle L,R,E \rangle , \mathbf{m},f)$ be an $\mathbf{m}$-labeled bipartite graph such that either L or R are finite. Then $\varGamma ^f$ is $\aleph _0$-categorical.

Proof

Without loss of generality assume that $L=\{l_1,l_2,\ldots , l_r\}$ is finite. Define a relation $\tau$ on R by $y \, \tau \, y'$ if and only if y and $y'$ are adjacent to the same elements in L and $\{l_i,y\}f = \{l_i,y'\}f$ for each such $l_i\in L$. Note that since both L and m are finite, R has finitely many $\tau$-classes, say $R_1,\ldots ,R_t$. Considering R simply as a set, fix ${\mathcal {A}}=(R;R_1,\ldots ,R_t)$.

Since L is finite, to prove the $\aleph _0$-categoricity of $\varGamma ^f$ it suffices to show that $(\varGamma ^f{\setminus } L)^n= R^n$ has finitely many $\sim _{\varGamma ^f,n}$-classes for each $n\in {{\mathbb {N}}}$ by a simple generalization of Gould and Quinn-Gregson [9, Proposition 2.11]. Let ${{\overline{a}}}=(r_1,\ldots ,r_n)$ and ${{\overline{b}}}=(r_1',\ldots ,r_n')$ be n-tuples of R such that ${{\overline{a}}} \, \sim _{{\mathcal {A}},n} \, {{\overline{b}}}$ via $\psi \in \text {Aut}({\mathcal {A}})$, say. We claim that the map ${{\hat{\psi }}}:\varGamma ^f\rightarrow \varGamma ^f$ which fixes L and is such that ${{\hat{\psi }}}|_R=\psi$ is an automorphism of $\varGamma ^f$. Indeed, as $\psi$ setwise fixes the $\tau$-classes, we have $(r,r\psi )\in \tau$ for each $r\in R$. Hence r and $r\psi$ are adjacent to the same elements in L, and so

$$\begin{aligned} \{l_i,r\}\in E \Leftrightarrow \{l_i,r\psi \} \in E \Leftrightarrow \{l_i{{\hat{\psi }}},r{{\hat{\psi }}}\}\in E, \end{aligned}$$

so that ${{\hat{\psi }}}$ is an automorphism of $\varGamma$. Similarly $\{l_i,r\}f=\{l_i,r\psi \}f=\{l_i{{\hat{\psi }}},r{{\hat{\psi }}}\}f$, so that ${{\hat{\psi }}}$ preserves labels. This proves the claim.

For each $1\le k \le n$ we have $r_k{{\hat{\psi }}}= r_k\psi =r_k'$, so that ${{\overline{a}}} \, \sim _{\varGamma ^f,n} \, {{\overline{b}}}$. Consequently,

$$\begin{aligned} |(\varGamma ^f{\setminus } L)^{n}/\sim _{\varGamma ^f,n}| \le |{\mathcal {A}}^n/\sim _{{\mathcal {A}},n}|. \end{aligned}$$

The set extension ${\mathcal {A}}$ is $\aleph _0$-categorical by Corollary 2.9, and so $|{\mathcal {A}}^n/\sim _{{\mathcal {A}},n}|$ is finite for each $n\ge 1$. Hence $\varGamma ^f$ is $\aleph _0$-categorical. $\square$

Lemma 5.23

Let $\varGamma ^f=(\langle L,R,E \rangle , \mathbf{m},f)$ be such that there exists $p\in \mathbf{m}$ with $\{x,y\}f=p$ for all but finitely many edges in $\varGamma$. Then $\varGamma ^f$ is $\aleph _0$-categorical if and only if $\varGamma$ is $\aleph _0$-categorical.

Proof

Suppose $\varGamma$ is $\aleph _0$-categorical, and that $\{l_1,r_1\},\ldots , \{l_t,r_t\}$ are precisely the edges of $\varGamma$ such that $\{l_k,r_k\}f\ne p$, where $l_k\in L$ and $r_k\in R$. Let ${{\overline{a}}}$ and ${{\overline{b}}}$ be n-tuples of $\varGamma ^f$ such that

$$\begin{aligned} ({{\overline{a}}},l_1,r_1,\ldots ,l_t,r_t) \, \sim _{\varGamma ,n+2t} \, ({{\overline{b}}},l_1,r_1,\ldots ,l_t,r_t) \end{aligned}$$

via $\psi \in \text {Aut}(\varGamma )$, say. We claim that $\psi$ is an automorphism of $\varGamma ^f$. For each $1\le k \le t$ we have $l_k\psi =l_k$ and $r_k\psi =r_k$ so that

$$\begin{aligned} \{l_k,r_k\}f= \{l_k\psi ,r_k\psi \}f. \end{aligned}$$

It follows that $\{l,r\}f=p$ if and only if $\{l\psi ,r\psi \}f=p$, and so $\psi$ preserves all labels, thus proving the claim. Consequently, ${{\overline{a}}} \, \sim _{\varGamma ^f,n} \, {{\overline{b}}}$ via $\psi$, so that

$$\begin{aligned} |(\varGamma ^f)^{n} /\sim _{\varGamma ^f,n}| \le |\varGamma ^{n+2t}/\sim _{\varGamma ,n+2t}|<\aleph _0 \end{aligned}$$

by the $\aleph _0$-categoricity of $\varGamma$. Hence $\varGamma ^f$ is $\aleph _0$-categorical.

The converse is immediate from Lemma 5.21. $\square$

Definition 5.24

Given a Rees matrix semigroup $S={\mathcal {M}}^0[G;I,\varLambda ;P]$, we form a G(P)-labeling of the induced bipartite graph $\varGamma (P) = \langle I,\varLambda , E\rangle$ of S in the natural way by taking the labeling $f:E\rightarrow G(P)$ given by

$$\begin{aligned} \{i,\lambda \}f = p_{\lambda ,i}. \end{aligned}$$

We denote the labeled bipartite graph by $\varGamma (P)^l$, which we call the induced labeled bipartite graph of S.

Note that, unlike the corresponding case for the induced bipartite graph $\varGamma (P)$, there exist isomorphic Rees matrix semigroups with non-isomorphic induced labeled bipartite graphs. For example, let G be a non-trivial group and P and Q be ${{\mathbf {1}}} \times {{\mathbf {2}}}$ matrices over $G\cup \{0\}$ given by

$$\begin{aligned} P = \left( \begin{array}{cc} 1&a \end{array} \right) \quad Q = \left( \begin{array}{cc} 1&1 \end{array} \right) \end{aligned}$$

where $a\notin \{0,1\}$. Let $S={\mathcal {M}}^0[G;{{\mathbf {2}}},{{\mathbf {1}}};P]$ and $T={\mathcal {M}}^0[G;{{\mathbf {2}}},{{\mathbf {1}}};Q]$, noting that $\varGamma (P)=\varGamma (Q)$ (and are isomorphic to $K_{2,1}$). Then $(1_G,1_{\varGamma (P)},(u_i)_{i\in {{\mathbf {2}}}},(v_{\lambda })_{\lambda \in {{\mathbf {1}}}})$ is an isomorphism from S to T, where $u_1=1=v_1$, and $u_2=a$. However, since $\varGamma (P)^l$ and $\varGamma (Q)^l$ have different labeling sets, they are not isomorphic.

Proposition 5.25

Let $S={\mathcal {M}}^0[G;I,\varLambda ;P]$ be a Rees matrix semigroup such that G and $\varGamma (P)^l$ are $\aleph _0$-categorical. Then S is $\aleph _0$-categorical.

Proof

Since $\varGamma (P)^l$ is $\aleph _0$-categorical, the set G(P) is finite by Lemma 5.21, say $G(P)=\{x_1,\ldots ,x_r\}$. Consider a pair of n-tuples ${{\overline{a}}}=((i_1,g_1,\lambda _1),\ldots , (i_n,g_n,\lambda _n))$ and ${{\overline{b}}}=((j_1,h_1,\mu _1),\ldots , (j_n,h_n,\mu _n))$ of $S^*$ under the pair of conditions that

• (1) $(g_1,\ldots ,g_n,x_1,\ldots ,x_r) \, \sim _{G,n+r} \, (h_1,\dots ,h_n,x_1,\ldots ,x_r)$,

• (2) $\varGamma ({{\overline{a}}}) \, \sim _{\varGamma (P)^l,2n} \, \varGamma ({{\overline{b}}})$,

via $\theta \in \text {Aut}(G)$ and $\psi \in \text {Aut}(\varGamma (P)^l)$, respectively (noting the use of Notation 5.9 here). We claim that $\phi =(\theta ,\psi ,(1)_{i\in I},(1)_{\lambda \in \varLambda })$ is an automorphism of S. Indeed, if $p_{\lambda ,i}\ne 0$ for some $i\in I,\lambda \in \varLambda$, then $p_{\lambda ,i}=x_k$ for some k, so that $\{i,\lambda \}f=\{i\psi ,\lambda \psi \}f=x_k$. Consequently,

$$\begin{aligned} p_{\lambda ,i}\theta =x_k\theta =x_k = p_{\lambda \psi ,i\psi }, \end{aligned}$$

and the claim follows by Theorem 5.10. Hence

$$\begin{aligned} (i_t,g_t,\lambda _t)\phi =(i_t\psi ,g_t\theta ,\lambda _t\psi )=(j_t,h_t,\mu _t) \end{aligned}$$

for each $1\le t \le n$, so that

$$\begin{aligned} |(S^*)^n/\sim _{S,n}|\le |G^{n+r}/\sim _{G,n+r}|\cdot |(\varGamma (P)^l)^{2n}/\sim _{\varGamma (P)^l,2n}|<\aleph _0, \end{aligned}$$

as G and $\varGamma (P)^l$ are $\aleph _0$-categorical. Hence S is $\aleph _0$-categorical by Gould and Quinn-Gregson [9, Proposition 2.11]. $\square$

The proposition above enables us to produce concrete examples of $\aleph _0$-categorical Rees matrix semigroups. For example, the result below is immediate from Lemma 5.22.

Corollary 5.26

Let S be a Rees matrix semigroup over an $\aleph _0$-categorical group having sandwich matrix P with finitely many rows or columns, and G(P) being finite. Then S is $\aleph _0$-categorical.

Similarly, Lemma 5.23 may be used in conjunction with Proposition 5.25 to obtain:

Corollary 5.27

Let $S={\mathcal {M}}^0[G;I,\varLambda ;P]$ be a Rees matrix semigroup such that G and $\varGamma (P)$ are $\aleph _0$-categorical, and all but finitely many of the non-zero entries of P are the identity of G. Then S is $\aleph _0$-categorical.

However, the converse to Proposition 5.25 fails to hold in general, and a counterexample will be constructed later in the next subsection. The idea is that any ${\mathcal {M}}^0[G;I,\varLambda ;P]$ in which G(P) is infinite forces $\varGamma (P)^l$ to be non $\aleph _0$-categorical by Lemma 5.25.

Open Problem 5.28

Does there exist an $\aleph _0$-categorical connected Rees matrix semigroup with G(P) finite which is not isomorphic to a Rees matrix semigroup with $\aleph _0$-categorical induced labeled bipartite graph?

We prove that the open problem has a negative answer for the case of completely simple semigroups. Given a completely simple semigroup ${\mathcal {M}}[G;I,\varLambda ;P]$, we call P normal if there exist $i\in I$ and $\lambda \in \varLambda$ such that $p_{\mu ,i}=p_{\lambda ,j}=1$ for every $j\in I$ and $\mu \in \varLambda$. Every completely simple semigroup is isomorphic to a Rees matrix semigroup without zero in which the sandwich matrix is normal [17].

Proposition 5.29

Let ${\mathcal {M}}[G;I,\varLambda ;P]$ be an $\aleph _0$-categorical completely simple semigroup in which P is normal and G(P) is finite. Then $\varGamma (P)^l$ is $\aleph _0$-categorical.

Proof

Suppose P is normalised via $i^*\in I$ and $\lambda ^*\in \varLambda$. Since G(P) is finite we may fix some finite subsets $I'=\{x_1,\ldots ,x_p\}\subseteq I$ and $\varLambda '=\{y_1,\ldots ,y_q\}\subseteq \varLambda$ such that the $\varLambda '\times I'$ submatrix of P contains every element of G(P). Let ${{\overline{x}}}$ be the pq-tuple of S given by

$$\begin{aligned} ((x_1,1,y_1),(x_1,1,y_2),\ldots ,(x_1,1,y_q), (x_2,1,y_1),\ldots ,(x_p,1,y_q)), \end{aligned}$$

Using the notation of Proposition 5.12, let ${{\overline{a}}}=(a_1,\ldots ,a_n)$ and ${{\overline{b}}}=(b_1,\ldots ,b_n)$ be a pair of $\sigma _{\varGamma (P)^l,n}$-related n-tuples of $\varGamma (P)^l$. Let $i_1<i_2<\cdots <i_s$ and $j_1<j_2<\cdots <j_t$ be the indexes of entries of ${{\overline{a}}}$ (and thus ${{\overline{b}}}$) lying in I and $\varLambda$, respectively. Suppose further that there exists $i\in I$ and $\lambda \in \varLambda$ such that the $n+pq+1$-tuples

$$\begin{aligned}&((a_{i_1},1,\lambda ),\ldots ,(a_{i_s},1,\lambda ),(i,1,a_{j_1}),\ldots ,(i,1,a_{j_t}), {{\overline{x}}},(i^*,1,\lambda ^*)) \text { and } \\&((b_{i_1},1,\lambda ),\ldots ,(b_{i_s},1,\lambda ),(i,1,b_{j_1}),\ldots ,(i,1,b_{j_t}), {{\overline{x}}},(i^*,1,\lambda ^*)) \end{aligned}$$

are automorphically equivalent via $\phi =[\theta ,\psi ,(u_i)_{i\in I},(v_{\lambda })_{\lambda \in \varLambda }] \in \text {Aut}(S)$, say. Then $\psi$ is an automorphism of $\varGamma (P)$ which maps ${{\overline{a}}}$ to ${{\overline{b}}}$. We aim to show that $\psi$ preserves labels, i.e., $p_{\lambda ,i}=p_{\lambda \psi ,i\psi }$ for every $i\in I, \lambda \in \varLambda$. Since $\phi$ fixes $(i^*,1,\lambda ^*)$ we have by Gould and Quinn-Gregson [24, Corollary 4.9] that there exists $g\in G$ with $u_i=g$ and $v_{\lambda }=g^{-1}$ for every $i\in I, \lambda \in \varLambda$. Since $\psi$ fixes $x_1,\ldots ,x_p,y_1,\ldots ,y_q$ we have

$$\begin{aligned} p_{y_k,x_{\ell }}\theta = v_{y_k} p_{y_k\psi , x_{\ell }\psi } u_{x_{\ell }} = g^{-1} p_{y_k,x_{\ell }} g. \end{aligned}$$

Consequently, as every $p_{\lambda ,i}$ is equal to some $p_{y_k,x_{\ell }}$, we have $p_{\lambda ,i}\theta = g^{-1} p_{\lambda ,i} g$ for every $i\in I$, $\lambda \in \varLambda$. However, $p_{\lambda ,i}\theta =v_\lambda p_{\lambda \psi ,i\psi }u_i=g^{-1} p_{\lambda \psi ,i\psi } g$, and hence $\psi$ preserves labels as required. We have thus shown that

$$\begin{aligned} (\varGamma (P)^l)^n / \sim _{\varGamma (P)^l,n}| \le |S^{n+pq+1}/\sim _{S, n+pq+1}| <\aleph _0 \end{aligned}$$

as S is $\aleph _0$-categorical. $\square$

The sandwich matrix P of a Rees matrix semigroup ${\mathcal {M}}^0[G;I,\varLambda ;P]$ can also always be normalised, but it is necessarily more complex. We can restate Open Problem 5.28 as follows:

Open Problem 5.30

If ${\mathcal {M}}^0[G;I,\varLambda ;P]$ is $\aleph _0$-categorical, where P is normal and G(P) is finite, then is $\varGamma (P)^l$ $\aleph _0$-categorical?

Notice that in Example 5.13, the labeled bipartite graph is clearly not $\aleph _0$-categorical since each $i_k$ is adjacent to exactly k vertices in which the edge is labeled by a. By construction the matrix P is normal via row $\alpha _0$ and column $i_0$, and hence S is not $\aleph _0$-categorical by the proposition above.

5.3 Pure completely 0-semigroups

Following Jackson and Volkov [18], we call a completely 0-simple semigroup S pure if it is isomorphic to a Rees matrix semigroup with sandwich matrix over $\{0,1\}$. Houghton [15] considered trivial cohomology classes of Rees matrix semigroups, a property which is proven in Sect. 2 of his article to be equivalent to being pure. Hence, by Houghton [15, Theorem 5.1], a completely 0-simple semigroup is pure if and only if, for each $a,b\in S$,

$$\begin{aligned} {[}a,b\in \langle E(S) \rangle \text { and } a \, {\mathcal {H}} \, b] \Rightarrow a=b. \end{aligned}$$

It follows that all orthodox completely 0-simple semigroups are necessarily pure, but the converse is not true in general. Indeed, a completely 0-simple semigroup is orthodox if and only if it is isomorphic to a Rees matrix semigroup with sandwich matrix over $\{0,1\}$ and with induced bipartite graph a disjoint union of complete bipartite graphs [13, Theorem 6]. Hence, in this case, it can be easily shown that the isomorphism types of the connected Rees components depends only on the isomorphism types of the induced (complete) bipartite graphs.

We observe that if the sandwich matrix of a Rees matrix semigroup is over $\{0,1\}$ then $\varGamma (P)^l$ is simply labeled by $\{1\}$. Therefore all automorphisms of $\varGamma (P)$ automatically preserve the labeling, and so $\varGamma (P)^l$ is $\aleph _0$-categorical if and only if $\varGamma (P)$ is $\aleph _0$-categorical. The equivalence of statements (1), (3), and (4) in the result below therefore follow from Propositions 5.12 and 5.25. For the interest of the reader we give an alternative proof of (4) $\Rightarrow$ (1) using results in [9].

Lemma 5.31

Let $S={\mathcal {M}}^0[G;I,\varLambda ;P]$ be a pure Rees matrix semigroup. Then the following are equivalent:

• (1) S is $\aleph _0$-categorical;

• (2) G and $\langle E(S)\rangle$ are $\aleph _0$-categorical;

• (3) G and $\varGamma (P)$ are $\aleph _0$-categorical;

• (4) G and ${\mathcal {M}}^0[\{1\};I,\varLambda ;P]$ are $\aleph _0$-categorical.

Proof

(1) $\Rightarrow$ (2) If S is $\aleph _0$-categorical then so is G by Proposition 5.12. Clearly E(S) is preserved by automorphisms of S, and hence $\langle E(S) \rangle$ is a characteristic subsemigroup of S, and thus inherits $\aleph _0$-categoricity.

(2) $\Rightarrow$ (3) Suppose that $\langle E(S) \rangle =\langle \{(i,1,\lambda ):p_{\lambda ,i}\ne 0\}\cup \{0\}\rangle$ is $\aleph _0$-categorical. Let $S_k={\mathcal {M}}^0[G;I_k,\varLambda _k;P_k]$ ($k\in A$) be the connected Rees components of S, where $P_k$ is the $\varLambda _k\times I_k$ submatrix of P. Then $\langle E(S) \rangle$ is isomorphic to the 0-direct union of the semigroups $E_k=\langle E(S_k) \rangle$, and since each $P_k$ is regular it is a simple exercise to show that $E_k = {\mathcal {M}}^0[\{1\};I_k,\varLambda _k;P_k]$. By Gould and Quinn-Gregson [9, Corollary 4.9] $\langle E(S) \rangle$ is $\aleph _0$-categorical if and only if each $E_k$ is $\aleph _0$-categorical and $\{E_k: k\in A\}$ is finite, up to isomorphism. By Proposition 5.12 each $\varGamma (P_k)$ is $\aleph _0$-categorical, and by Theorem 5.10${\mathcal {C}}(\varGamma (P))=\{\varGamma (P_k):k\in A\}$ is finite, up to isomorphism. Hence $\varGamma (P)$ is $\aleph _0$-categorical by Proposition 5.5.

(3) $\Rightarrow$ (4) Immediate from Corollary 5.27.

(4) $\Rightarrow$ (1) The elements of the combinatorial Rees matrix semigroup $T={\mathcal {M}}^0[\{1\};I,\varLambda ;P]$ can be identified² with the set $(I\times \varLambda )\cup \{0\}$. Since $\aleph _0$-categoricity is preserved by finite direct products [12], the semigroup $U=G \times T$ is $\aleph _0$-categorical. The set $I=\{(g,0):g\in G\}$ is an ideal of U, and the Rees quotient U/I is a principal factor of U. Hence U/I is $\aleph _0$-categorical by Gould and Quinn-Gregson [9, Theorem 3.12]. Moreover, the map $\phi :U/I\rightarrow S$ given by $0\phi =0$ and $(g,(i,\lambda ))\phi = (i,g,\lambda )$ ($g\in G, i\in I,\lambda \in \varLambda$) is an isomorphism, to which the result follows. $\square$

Furthermore, since complete bipartite graphs are $\aleph _0$-categorical by Theorem 5.3, a disjoint union of complete bipartite graphs is $\aleph _0$-categorical if and only if it has finitely many connected components, up to isomorphism, by Proposition 5.5. The corollary above thus reduces in the orthodox case as follows.

Corollary 5.32

Let $S={\mathcal {M}}^0[G;I,\varLambda ;P]$ be an orthodox Rees matrix semigroup. Then the following are equivalent:

• (1) S is $\aleph _0$-categorical;

• (2) G and E(S) are $\aleph _0$-categorical;

• (3) G is $\aleph _0$-categorical and $\varGamma (P)$ has finitely many connected components, up to isomorphism;

• (4) G and ${\mathcal {M}}^0[\{1\};I,\varLambda ;P]$ are $\aleph _0$-categorical.

In [9] we studied inverse completely 0-simple semigroups, that is, Brandt semigroups. These are necessarily orthodox, and are isomorphic to a Rees matrix semigroup of the form ${\mathcal {M}}^0[G;I,I;P]$ where P is the identity matrix, that is, $p_{ii}=1$ and $p_{ij}=0$ for each $i\ne j$ in I, and are denoted ${\mathcal {B}}^0[G;I]$. Since the induced biparite graph of a Brandt semigroup is a perfect matching, it is $\aleph _0$-categorical by Theorem 5.3. Corollary 5.32 then simplifies to obtain our classification of $\aleph _0$-categorical Brandt semigroups [9, Theorem 4.2], which states that a Brandt semigroup over a group G is $\aleph _0$-categorical if and only if G is $\aleph _0$-categorical.

We are now able to construct a simple counterexample to the converse of Proposition 5.25. Let $G=\{g_i:i\in {{\mathbb {N}}}\}$ be an infinite $\aleph _0$-categorical group. Let

$$\begin{aligned} S={\mathcal {M}}^0[G;{{\mathbb {N}}},{{\mathbb {N}}};P]={\mathcal {B}}^0[G;{{\mathbb {N}}}] \text { and } T={\mathcal {M}}^0[G;{{\mathbb {N}}},{{\mathbb {N}}};Q], \end{aligned}$$

where $Q=(q_{i,j})$ is such that $q_{i,i}=g_i$ and $q_{i,j}=0$ for each $i\ne j$. Then $\varGamma (P)=\varGamma (Q)$ (and are isomorphic to $P_{{{\mathbb {N}}}}$) and $(1_G,1_{\varGamma (P)},(g^{-1}_i)_{i\in {{\mathbb {N}}}},(1)_{\lambda \in {{\mathbb {N}}}})$ is an isomorphism from S to T by Theorem 5.10 since

$$\begin{aligned} p_{i,i}1_G=1=g_ig^{-1}_i = 1 \cdot q_{i,i} \cdot g^{-1}_i, \end{aligned}$$

for each $i\in {{\mathbb {N}}}$. Since S is $\aleph _0$-categorical by the $\aleph _0$-categoricity of G, the same is true of T. However, $\varGamma (Q)^l$ is a G-labeling, and is thus not $\aleph _0$-categorical by Lemma 5.21. Hence T is our desired counterexample.

5.4 Alternative directions

To further incorporate the link between the induced bipartite graph of a Rees matrix semigroup and the entries of the sandwich matrix, we could instead introduce the stronger notion of an induced group labeled bipartite graph. A group labeled bipartite graph is a G-labeled bipartite graph $\varGamma ^f=(\langle L,R,E \rangle ,G,f)$, for some group G, where an automorphism of $\varGamma ^f$ is a pair $(\psi ,\theta )\in \text {Aut}(\varGamma )\times \text {Aut}(G)$ such that, for each $\ell \in L, r\in R$,

$$\begin{aligned} (\ell ,r)f=g \Leftrightarrow (\ell \psi ,r\psi )f=g\theta . \end{aligned}$$

However, group labeled biparite graphs do not appear to be first-order structures.

Let $S={\mathcal {M}}^0[G;I,\varLambda ;P]$ be such that G(P) forms a subgroup of G. Then we may define the induced group labeled bipartite graph of S as the G(P)-labeled bipartite graph $\varGamma (P)^f$, with automorphisms being pairs $(\psi ,\theta )\in \text {Aut}(\varGamma ) \times \text {Aut}(G(P))$ such that $p_{\lambda \psi ,i\psi }=p_{\lambda ,i}\theta$ for each $i\in I, \lambda \in \varLambda$. Notice that if $(\psi ,\theta )$ is an automorphism of the induced group labeled bipartite graph of S and is such that $\theta$ extends to an automorphism $\theta '$ of G, then $(\theta ',\psi ,(1)_{i\in I},(1)_{\lambda \in \varLambda })$ is clearly an automorphism of S. However, we do not in general obtain all automorphisms of S in this way. Similar problems therefore arise in regard to when $\aleph _0$-categoricity of S passes to its induced group labeled bipartite graph (by which we mean the induced group labeled bipartite graph has an oligomorphic automorphism group).

An alternative next step could be to extend the scope of this section by considering the $\aleph _0$-categoricity of Rees matrix semigroups over semigroups (or monoids), denoted ${\mathcal {M}}^0[S;I,\varLambda ;P]$, where again we assume P is regular. Similarly we may define ${\mathcal {M}}[S;I,\varLambda ;P]$. However, this task is as difficult as considering the $\aleph _0$-categoricity of all semigroups. Indeed, if S is a semigroup then $T={\mathcal {M}}^0[S^1;\{i\},\{\lambda \};(1)]$ is isomorphic to S with both a zero and an identity adjoined, and by Gould and Quinn-Gregson [9, Corollary 2.12] S is $\aleph _0$-categorical if and only if T is $\aleph _0$-categorical. A second problem that arises is that the vital Theorem 5.10 only holds in the forwards direction for Rees matrix semigroups over semigroups. As such we do not have an explicit description of the automorphism group of ${\mathcal {M}}^0[S;I,\varLambda ;P]$ via its components, and many of the proofs of this section do not seem to be easily extendable. In fact the $\aleph _0$-categoricity of a Rees matrix semigroup over a semigroup S does not necessarily pass to S, unlike for groups as shown in Proposition 5.12. For example, take any semigroup S with zero element $\epsilon$, and consider $M={\mathcal {M}}[S;\{i\},\{\lambda \};(\epsilon )]$. Then M is isomorphic to a null semigroup with zero element $(i,\epsilon ,\lambda )$, which is $\aleph _0$-categorical by Gould and Quinn-Gregson [9, Example 2.7]; taking S to be non $\aleph _0$-categorical gives our desired example. On the other hand, it can be easily shown that Proposition 5.25 can be extended to Rees matrix semigroups over monoids. This allows us to build chains of $\aleph _0$-categorical semigroups as follows. Let M be an $\aleph _0$-categorical monoid, and let P be a $\varLambda \times I$ matrix over $\{0,1\}$ in which $\varGamma (P)$ is $\aleph _0$-categorical. Take $M_1={\mathcal {M}}^0[M;I,\varLambda ;P]$, and inductively define $M_k={\mathcal {M}}^0[M_{k-1}^1;I,\varLambda ;P]$ for $k>1$. Then each $M_k$ is $\aleph _0$-categorical, and $M_{k-1}$ embeds into $M_k$, for each $k\in {{\mathbb {N}}}$.