On the automorphism groups of vertex-transitive Cayley digraphs of monoids

Abstract

In 1982, Babai and Godsil conjectured that almost all Cayley digraphs are digraphical regular representations. In 1998, Xu conjectured that almost all Cayley digraphs are normal [i.e., $G_L$ is a normal subgroup of the automorphism group of $\text {Cay}(G,C)$]. Finally, in 1994, Praeger and Mckay conjectured that almost all undirected vertex-transitive graphs are Cayley graphs of groups. In this paper, first we present the variants of these conjectures for Cayley digraphs of monoids and we determine when the variant of Babai and Godsil’s conjecture is equivalent to the variant of Xu’s conjecture. Then, as a special consequence of our results, we conclude that Xu’s conjecture is equivalent to Babai and Godsil’s conjecture. On the other hand, we give affirmative answer to the variant of Praeger and Mckay’s conjecture and we prove that a Cayley digraph of a monoid is vertex-transitive if and only if it is isomorphic to a Cayley digraph of a group. Finally, we use this characterization to give an affirmative answer to a question raised by Kelarev and Praeger about vertex-transitivity of Cayley digraphs of monoids. Also using this characterization, we explicitly determine the automorphism groups of vertex-transitive Cayley digraphs of monoids.

1 Introduction and preliminaries

Throughout this paper, all semigroups are finite semigroups. Let C be a non-empty subset of a semigroup S. The Cayley digraph $\text {Cay}(S,C)$ of S with respect to C is defined as the digraph with vertex set S and arc set $E(\text {Cay}(S,C))$ consisting of those ordered pairs $(s, s')$ such that $sc=s'$, for some $c \in C$.

The main goal of this paper is to investigate the automorphism groups of Cayley digraphs of monoids. For this purpose, we use the known results about Cayley digraphs of groups and we study the variants of some well-known conjectures about them (Problems 2.3, 2.8 and 2.9). To illustrate the application of our results, we study Cayley digraphs of groups. In Theorem 1.7, we prove that Xu’s conjecture (Conjecture 1.2) is equivalent to Babai and Godsil’s conjecture (Conjecture 1.1). Also in Proposition 4.4, we give affirmative answer to the variant of Praeger and Mckay’s conjecture for Cayley digraphs of monoids (see Conjecture 1.3 and Problem 1.5). Then, we use this result to give affirmative answer to Problem 1.4 for Cayley digraphs of monoids (see Proposition 4.6). Furthermore, using Theorem 4.3 and Proposition 4.4, we determine the automorphism groups of vertex-transitive Cayley digraphs of monoids explicitly (see Proposition 4.7). Finally, we state the origin of our approach, which is based on topology (see Remark 4.9).

For every monoid S, the identity element of S is denoted by $1_S$ and simply by 1 if there is no ambitious about S. By U(S), we mean the set of invertible elements of S. Let $C\subseteq S$. We denote $\mathrm {End}(\text {Cay}(S,C))$ by $\mathrm {End}_{C}(S)$, and $\text {Aut}(\text {Cay}(S,C))$ by $\text {Aut}_{C}(S)$. Recall that for every $u\in U(S)$, $\lambda _u:S\rightarrow S$ defined by $s\mapsto us$ belongs to $\text {Aut}_C(S)$. We denote $\{\lambda _u\mid u\in U(S)\}$ by $U(S)_L$. A bijection $\phi :S\rightarrow S$ is called a monoid automorphism if for every $s,s'\in S$ we have $\phi (ss')=\phi (s)\phi (s')$ and $\phi (1)=1$. We denote the automorphism group of S by $\text {Aut}(S)$. The identity function on a monoid S is denoted by $\text {id}_S$.

Recall that for a group G and a subset $C\subseteq G$, the Cayley digraph $\text {Cay}(G,C)$ is a digraphical regular representation (DRR) of G if $\text {Aut}_C(G) = G_L$, the left regular representation of G. A Cayley digraph $\text {Cay}(G,C)$ is normal, if $G_L$ is a normal subgroup of $\text {Aut}_C(G)$. Similarly, recall that for a monoid S and $C\subseteq S$, the Cayley digraph $\text {Cay}(S,C)$ is called normal, if $\text {ColAut}_C(S)$ is a normal subgroup of $\text {Aut}_C(S)$ (see [20]). Note that there is also another definition of normal Cayley digraphs of groups which is proposed by Praeger in [36]. However, since her notion of normality is based on edge-transitivity of Cayley digraphs of groups, the variant of her notion of normality for Cayley digraphs of monoids is trivial and so we do not consider it (see [20, Remark 3.16]). Now note that if a Cayley digraph of a group is a DRR, then it is normal. Recall that in [2], Babai and Godsil suggested the following conjecture.

Conjecture 1.1

[4, Conjecture 1.1] Let G be a group of order n. The proportion of subsets C of G such that $\text {Cay}(G,C)$ is a DRR goes to 1 as $n\rightarrow \infty$.

First, note that this conjecture is about Cayley digraphs of groups, and it is not true for undirected Cayley graphs of groups, because for every abelian group G of odd order, the function $\iota :G\rightarrow G$ defined by $\iota (g)=-g$, belongs to $\text {Aut}_C(G)\setminus G_L$ (see [4]). Now recall that as a main result in [2], Babai and Godsil proved this conjecture for nilpotent groups of odd order. Also in [5] it was proved that when G is a p-group with no homomorphism onto the wreath product of ${\mathbb {Z}}_p$ by ${\mathbb {Z}}_p$, almost all Cayley digraphs of G have automorphism group isomorphic to G. On the other hand, Xu in [39] proposed the following conjecture.

Conjecture 1.2

[39, Conjecture 1] Almost every Cayley digraphs of groups are normal.

Later in [4] Dobson et al. proved these conjectures for abelian groups. Recall that in [5, Lemma 2.2] or [39, Proposition 1.3] it was proved that $\text {Cay}(G,C)$ is normal if and only if $\text {Aut}_C(G)$ is isomorphic to $G_L\rtimes \text {Aut}(G,C)$, where $\text {Aut}(G,C)=\{\sigma \in \text {Aut}(G)\mid \sigma (C)=C\}$. Note that for a group G and a subset C of G, if $A_1$ denotes the stabilizer in $\text {Aut}_C(G)$ of the identity element of G, then the cases where $\text {Aut}(G,C)=\{\text {id}_G\}$ and $A_1\ne \{\text {id}_G\}$, are believed to be very rare, and for many interesting cases it is indeed true that $\text {Aut}(G,C)=\{\text {id}_G\}$ if and only if $A_1=\{\text {id}_G\}$ (see [32]). However, note that there are groups G with some generating sets C of G such that $\text {Aut}_C(G)\setminus (G_L \rtimes \text {Aut}(G,C))\ne \emptyset$. Also note that there are examples of groups G and some generating sets $C\subseteq G$ such that $\text {Aut}(G,C)=1$ and $\text {Cay}(G,C)$ is not a DRR (see [9]).

Now we proceed to state Praeger and Mckay’s conjecture and its variant for Cayley digraphs of monoids. Recall that a digraph is called a group digraph, if it is isomorphic to a Cayley digraph of some group. Similarly, for undirected graphs, we can define group graphs. Recall that a digraph $\Gamma$ is vertex-transitive, if for every two vertices u and v in $\Gamma$, there exists $\sigma \in \text {Aut}(\Gamma )$ such that $\sigma (u)=v$. Similarly, the digraph $\Gamma$ is called arc-transitive if for every pair of arcs $e,e'\in \Gamma$, there exists $\sigma \in \text {Aut}(\Gamma )$ such that $\sigma (e)=e'$. Note that for every group G and a subset C of G, the Cayley digraph $\text {Cay}(G,C)$ is vertex-transitive. Now recall that Praeger and Mckay suggested the following conjecture about undirected vertex-transitive graphs.

Conjecture 1.3

([1] or [35]) Almost all undirected vertex-transitive graphs are Cayley graphs of groups.

On the other hand, note that the Cayley digraphs of monoids and semigroups are not necessarily vertex-transitive (see [16, Example 3] or [23, Proposition 3.14]) and this gives a motivation for presenting several notions of vertex-transitivity of Cayley digraphs of semigroups and monoids. In the following, we recall some of these notions and facts about them. For more information about this topic, we refer to [14, 16] or [30].

Recall that for a semigroup S and a subset C of S, an element $\sigma \in \mathrm {End}_{C}(S)$ is called a color-preserving endomorphism if $xc=y$ implies $\sigma (x)c=\sigma (y)$, for every $x,y\in S$ and $c\in C$. The set of all color-preserving endomorphisms of $\text {Cay}(S,C)$ is denoted by $\mathrm {ColEnd}_{C}(S)$. The set of all color-preserving automorphisms of $\text {Cay}(S,C)$ is also denoted by $\text {ColAut}_C(S)$ (see [16]). A digraph automorphism $\sigma$ of $\text {Cay}(S,C)$ is called a color-permutable automorphism of $\text {Cay}(S,C)$, if there exists $\nu \in \mathrm {Sym}(C)$ such that for every $s\in S$ and $c\in C$, $\sigma (sc)=\sigma (s)\nu (c)$. Clearly, the set of all color-permutable automorphisms of $\text {Cay}(S,C)$ forms a subgroup of $\text {Aut}_C(S)$. We denote this subgroup by $\text {CPAut}_C(S)$ (see [20]). Let S be a monoid and $C\subseteq S$. Let $\text {Aut}(S,C)$ be the subgroup of $\text {Aut}(S)$ consisting of monoid automorphisms $\sigma$ such that $\sigma (C)=C$. We say the Cayley digraph $\text {Cay}(S,C)$ has regular affine automorphism group, if $|\text {Aut}(S,C)|=1$.

The Cayley digraph $\text {Cay}(S,C)$ is said to be color-vertex-transitive or $\text {ColAut}_{C}(S)$ -vertex-transitive, if for every two vertices $x,y\in S$, there exists some $f\in \text {ColAut}_{C}(S)$ such that $f(x)=y$. The notion of color-permutable vertex-transitive or $\text {CPAut}_{C}(S)$ -vertex-transitive for Cayley digraphs of semigroups is defined similarly (for some known results about this topic, see [10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28, 30, 33, 34, 37] or [38]). Kelarev and Praeger in [16] remarked that for every semigroup S and a subset C of S we have

$$\begin{aligned} \text {Cay}(S,C)=\cup _{c\in C}\text {Cay}(S,\{c\}). \end{aligned}$$

Then, based on this fact, they raised the following problem.

Problem 1.4

[16, Question 1] Is it true that if S is a semigroup with a subset C such that $\text {Cay}(S, \{c\})$ is $\text {Aut}_{\{c\}}(S)$-vertex-transitive, for every $c\in C$, then the whole Cayley digraph $\text {Cay}(S,C)$ is $\text {ColAut}_C(S)$-vertex-transitive, too?

Later in [10, Example], Jiang showed that the answer to this question is negative, in general (his example is a semigroup which is not a monoid). So it is very logical and interesting to determine under which conditions, the answer to this question is affirmative.

In [19, Theorem 3.7], it was proved that for a finite semigroup S and $C\subseteq S$, if the Cayley digraph $\text {Cay}(S,C)$ is $\text {ColAut}_C(S)$-vertex-transitive, then $\text {Cay}(S,C)$ is a group digraph. Later it was proved that a finite Cayley digraph $\text {Cay}(S,C)$ is $\mathrm {ColEnd}_C(S)$-vertex-transitive, where C is a generating set of a monoid S if and only if S is a group (see [20, Theorem 3.4]). Based on these results, it may seem that a Cayley digraph of a monoid is color-vertex-transitive if and only if it is a group digraph. However, note that there is an infinite monoid S with some element $c\in S$ such that $\text {Cay}(S,\{c\})$ is color-vertex-transitive and $\text {Cay}(S,\{c\})$ is not a group digraph (see [27, Theorem 3.1]). On the other hand, note that there is an infinite color-permutable vertex-transitive Cayley digraph of some monoid which is not color-vertex-transitive (see [20, Example 3.3]). Finally, note that there is a finite semigroup S which is not a monoid (group), and a subset $C\subseteq S$ such that $\text {Cay}(S,C)$ is vertex-transitive (see Example 4.1). Excitingly, all known vertex-transitive Cayley digraphs of finite monoids are group digraphs (see, for example, [19, 20] or [25]).

Based on the above discussions, it is very reasonable to consider Praeger and Mckay’s conjecture for Cayley digraphs of semigroups and specially monoids. Clearly, the variant of this conjecture for Cayley digraphs of semigroups is as follows:

Problem 1.5

Almost all vertex-transitive Cayley digraphs of semigroups (monoids) are Cayley digraphs of groups.

Note that if a digraph $\Gamma$ is vertex-transitive, then the underlying undirected graph $\Gamma$ is vertex-transitive, too (note that if $\sigma$ is an automorphism of $\Gamma$, then $\sigma$ is also an automorphism of the underlying undirected graph of $\Gamma$). So every answer to Problem 1.5 leads to an answer to Conjecture 1.3. In fact, if an undirected graph $\Gamma$ is the underlying undirected graph of a Cayley digraph of a monoid which is vertex-transitive, then $\Gamma$ is isomorphic to a Cayley graph of a group.

Now we recall some facts and notions, we need in the sequel. Recall that a digraph $\Gamma$ is called connected if its underlying undirected graph is connected. A digraph $\Gamma$ is called strongly connected if for every $u, v\in V (\Gamma )$, there exists a directed path from u to v.

For every set X, by |X| we mean the cardinality of X and by ${\mathbb {P}}(X)$ we mean the power set of X. We denote the family of all finite groups by $FGrp$. For every monoid S, by $S^1$ we mean the semigroup S with an adjoined identity 1. Let S be a monoid without zero and $m\in {\mathbb {N}}$. Let $[m] = \{1, 2,\ldots ,m\}$ and $B(S, m) = ([m]\times S \times [m]) \cup \{0\}$. Now define a binary operation (say, multiplication) on B(S, m) by

$$\begin{aligned} (i, a, j)(k, b, l)= & {} {\left\{ \begin{array}{ll} {(i, ab, l),} &{}\quad \text {if } j = k;\\ \text {0,} &{}\quad \text {otherwise.} \\ \end{array}\right. }\\ \text{ and } 0(i, a, j)= & {} (i, a, j)0 = 00 = 0. \end{aligned}$$

With the above defined multiplication, B(S, m) is a semigroup known as a Brandt semigroup (see [31]). For every $m\in {\mathbb {N}}$, by ${{\mathcal {B}}}{{\mathcal {S}}}(m)$ we mean $\{B(G,m)\mid G\in FGrp \}$. Note that ${{\mathcal {B}}}{{\mathcal {S}}}(1)= FGrp$. Similarly, by ${{\mathcal {B}}}{{\mathcal {S}}}^1(m)$ we mean $\{B^1(G,m)\mid G\in FGrp \}$.

The rank of a finite semigroup S, which is usually denoted by $\text {rank}(S)$, is the smallest number of elements required to generate S (see [6, 8] or [22]). The large rank of a finite semigroup S, denoted by $r_5(S)$, is the least number n such that every subset of S with n elements generates S (see [8]). The following theorem determines the large ranks of finite semigroups.

Theorem 1.6

[8, Theorem 3] Let S be a finite semigroup and W the largest proper subsemigroup of S. Then, $r_5(S) =|W|+1$.

As one of the main results in this paper, for a class of monoids ${\mathcal {F}}$, we use ranks, large ranks and automorphism groups of monoids in ${\mathcal {F}}$, to present a necessary condition under which almost all Cayley digraphs in ${\mathcal {F}}$ have regular affine automorphism groups (see Theorem 2.11). As a special consequence of using this theorem, we prove the following result.

Theorem 1.7

Xu’s conjecture is equivalent to Babai and Godsil’s conjecture.

Then, we continue our study for determining the automorphism groups of Cayley digraphs of semigroups in [20]. For this purpose, we give affirmative answer to Problem 1.5 and we show that every Cayley digraph of a monoid is a group digraph. Using this result, we connect the study of the automorphism groups of Cayley digraphs of monoids to the automorphism groups of group digraphs which are extensively studied (see, for example, [1, 2, 4, 5, 9, 32] and [39]).

2 Main results

Before we state our main results, we recall a very well-known concept in graph theory. Denote by D(n) the set of all digraphs without multiple arcs but possibly with some loops and with n vertices, and denote by DP(n) the set of all digraphs from D(n) with a certain property P. We say that almost all digraphs have property P if

$$\begin{aligned} \lim _{n\rightarrow \infty }{\mathrm{DP}(n)\over D(n)}=1 \end{aligned}$$

(for more details, see [30]).

In this note, we present the following notion of normality which is closely related to Xu’s notion of normality for Cayley digraphs of groups, too. In Sect. 4, we will study the relation between normal Cayley digraphs of monoids and this notion of normality (see Theorem 4.8). Recall that if for a group G, a Cayley digraph $\text {Cay}(G,C)$ is normal, then either $\text {Aut}_C(G)=G_L$, or $G_L\lvertneqq \text {Aut}_C(G)$ and $|\text {Aut}(G,C)|\ne 1$.

Definition 2.1

For a monoid S and a subset C of S, we say $\text {Cay}(S,C)$ is U-normal, if the following conditions hold:

• $U(S)_L$ is a normal subgroup of $\text {Aut}_C(S)$;

• if $U(S)_L\lvertneqq \text {Aut}_C(S)$, then $|\text {Aut}(S,C)|\ne 1$.

Clearly, by [20, Lemma 3.9], if $\text {Aut}_C(S)=\text {CPAut}_C(S)$, then $\text {Cay}(S,C)$ is normal and U-normal (for an example of such a monoid, see [20, Example 3.17]).

To simplify our discussion, we present the following notations.

Notation 2.2

For every natural number $n\in {\mathbb {N}}$, by ${\mathcal {M}}_n$ and ${\mathcal {G}}_n$, we mean the collection of monoids and groups of order n, respectively.

For every monoid S, let

$$\begin{aligned} bg (S)=\{C\subseteq S\mid \text {Aut}_C(S)=U(S)_L\}, \end{aligned}$$

and

$$\begin{aligned} \mathrm{Norm} (S)=\{C \subseteq S\mid \text {ColAut}_C(S)\unlhd \text {Aut}_C(S)\}, \end{aligned}$$

and

$$\begin{aligned} \mathrm{Norm} _U(S)=\{C \subseteq S\mid \text {Cay}(S,C)\,\,\text{ is }\,\,U\text{-normal }\}. \end{aligned}$$

For every subset C of S, note that $U(S)_L$ is a subgroup of $\text {ColAut}_C(S)$ and if C is a generating set of S, then $\text {ColAut}_C(S)=U(S)_L$ (see Lemma 2.5). Note that if C is not a generating set of S, then $U(S)_L$ is not necessarily equal to $\text {ColAut}_C(S)$ [see Example 2.6 (i)]. Furthermore, if $U(S)_L=\text {ColAut}_C(S)$ and $\text {Cay}(S,C)$ is connected, then C is not necessarily a generating set of S [see Example 2.6 (ii)].

Now note that for every $C\in bg (S)$, we have $\text {Aut}_C(S)=\text {ColAut}_C(S)=U(S)_L$. Hence,

$$\begin{aligned} bg (S)\subseteq \mathrm{Norm} _U(S)\cap \mathrm{Norm} (S). \end{aligned}$$

Let ${\mathcal {P}}(S)$ be the set of minimal subgroups of $\text {Aut}(S)$ (equivalently, the set of subgroups of $\text {Aut}(S)$ which have prime orders). For every $P\in {\mathcal {P}}(S)$, we denote the set of orbits of P on S by $\beta (P)$ (it is easy to see that $\beta (P)$ is a Boolean algebra. We use this fact in Remark 4.9). Let

$$\begin{aligned} {\mathcal {O}}(S)=\{\beta (P)\mid P\in {\mathcal {P}}(S)\}. \end{aligned}$$

Now it is easy to show that Conjecture 1.1 is equivalent to the following condition:

$$\begin{aligned} \lim _{n\rightarrow \infty }\min _{G\in {\mathcal {G}}_n} {|bg (G)|\over 2^{|G|}}=1. \end{aligned}$$

Similarly, Xu’s conjecture can be stated as follows:

$$\begin{aligned} \lim _{n\rightarrow \infty }\min _{G\in {\mathcal {G}}_n} {|\mathrm{Norm} _U(G)|\over 2^{|G|}}=1. \end{aligned}$$

It is clear that if Babai and Godsil’s conjecture is true, then Xu’s conjecture is true, too.

Based on these conjectures, in [20], the following problem (which is a variant of Xu’s conjecture for Cayley digraphs of monoids) is presented.

Problem 2.3

[20, Problem 3.18] Is

$$\begin{aligned} \lim _{n\rightarrow \infty }\min _{S\in {\mathcal {M}}_n} {\text{ the } \text{ number } \text{ of } \text{ normal } \text{ Cayley } \text{ digraphs } \text{ of } S\over \text{ the } \text{ number } \text{ of } \text{ Cayley } \text{ digraphs } \text{ of } S}=1? \end{aligned}$$

For every automorphism $h\in \text {Aut}(S)$ by $\text {Fix}(h)$ we mean the set of fixed points of h in S. Similarly, for every subgroup $H\le \text {Aut}(S)$, let $\text {Fix}(H)=\cap _{h\in H} \text {Fix}(h)$. Let $\text {NFix}(h)=S\setminus \text {Fix}(h)$ and $\text {NFix}(H)=S\setminus \text {Fix}(H)$. Note that if |H| is a prime number, then $\text {Fix}(H)=\text {Fix}(h)$ for every $h\in H\setminus \{\text {id}_S\}$. Clearly, $\text {Fix}(H)$ and $\text {Fix}(h)$ are subsemigroups of S. Also note that $\text {Fix}(H):=\{s\in S\mid H\cdot \{s\}=\{s\}\}$. The next remark shows the relation between $|S|-|\text {Fix}(h)|$ and large rank of S.

Remark 2.4

First, note that for every monoid S and an automorphism $\sigma \in \text {Aut}(S)$, if the order of $\sigma$ is a prime number p, then by Burnside’s theorem we have $p\mid |S|-|\text {Fix}(P)|$. Since in our study, we need to use the cardinality of $\text {NFix}(\sigma )$ (see Corollary 3.9), we use the large rank of S to present an upper bound of this cardinality. For this purpose, note that there does not exist any generating set C of S such that $C\subseteq \text {Fix}(\sigma )$. Therefore, by the definition of $r_5(S)$ we have $|\text {Fix}(\sigma )|<r_5(S)$.

Note that for every subset C of a monoid S, we have $U(S)_L\le \text {Aut}_C(S)$. However, we have the following better result when C is a generating set.

Lemma 2.5

[20, Corollary 2.4] Let S be a monoid and C be a generating set of S. Then $\text {ColAut}_C(S)=U(S)_L$.

The next example shows that $U(S)_L$ is not necessarily equal to $\text {ColAut}_C(S)$, where S is a monoid and $C\subseteq S$. Also it shows that if $\text {ColAut}_C(S)=U(S)_L$ and $\text {Cay}(S,C)$ is connected, then C is not necessarily a generating set of S.

Example 2.6

• (i) Let $G=({\mathbb {Z}}_4,+)$ and $C=\{{\overline{2}}\}$. Then, the function $\sigma :G\rightarrow G$ defined by $\sigma ({\overline{2}})={\overline{1}}$, $\sigma ({\overline{0}})={\overline{3}}$, $\sigma ({\overline{3}})={\overline{0}}$ and $\sigma ({\overline{1}})={\overline{2}}$, belongs to $\text {Aut}_C(G)$. On the other hand, since $|C|=1$, we have $|\text {Aut}_C(G)|=|\text {ColAut}_C(G)|$. We show that $\sigma \in \text {ColAut}_C(G)\setminus G_L$. Suppose to the contrary that $\sigma \in G_L$ and $\sigma =\lambda _g$. Then $g+{\overline{2}}={\overline{1}}$ and $g+{\overline{3}}={\overline{0}}$. So $g={\overline{3}}$ and $g={\overline{1}}$, which is a contradiction. Therefore, by the above discussion, $G_L\ne \text {ColAut}_C(G)$.

• (ii) Suppose that R is a right-zero semigroup and $|R|>1$. Let $S=R^1$ be the semigroup R with adjoined identity 1. Note that $U(S)=\{1\}$. We show that $\text {ColAut}_R(S)=U(S)_L=\{id_S\}$. Let $r_0\in R$. Note that for every $r\in R$, there exists an arc from r to $r_0$ and the only loop with color $r_0$ is incident to the vertex $r_0$. On the other hand, since the out-degree of 1 is |R|, for every $\sigma \in \text {Aut}_R(S)$ we have $\sigma (1)=1$. So for every $\sigma \in \text {ColAut}_R(S)$ we have $\sigma =id_S$ and therefore $\text {ColAut}_R(S)=U(S)_L=\{id_S\}$.

Now note that $U(S)_L\rtimes \text {Aut}(S,C)\le \text {CPAut}_C(S)$ and we have the following better result for generating sets.

Theorem 2.7

[20, Theorem 1.1] Let S be a monoid and C be a generating set of S. Then $\text {CPAut}_C(S)=U(S)_L\rtimes \text {Aut}(S,C)$.

Based on the results about Cayley digraphs of groups, it is reasonable to consider the following problem which is in fact the variant of Babai and Godsil’s conjecture for Cayley digraphs of monoids.

Problem 2.8

For almost all Cayley digraphs of monoids $\text {Cay}(S,C)$, is $\text {Aut}_C(S)=U(S)_L$?

For a monoid S and $C\subseteq S$, clearly $\text {Aut}_C(S)$ acts naturally on S (for a good resource about actions, see [29]). From now on, for every $\sigma \in \text {Aut}_C(S)$ and $s\in S$, by $\sigma \cdot s$ we mean the natural action of $\sigma$ on s defined by $\sigma \cdot s=\sigma (s)$. Similarly, for every $P\in {\mathcal {P}}(S)$ and $C\subseteq S$, let $P\cdot C=\{\sigma (c)\mid \sigma \in \ P,\ c\in C\}$.

For a monoid S and a subgroup $Q\le \text {Aut}(S)$, by N(Q) we mean the following set

$$\begin{aligned} N(Q)=\{C\subseteq S\mid Q\cdot C=C\}. \end{aligned}$$

Also let

$$\begin{aligned} {\mathcal {T}}(S)= & {} \{C\subseteq S\ \big \vert \ |\text {Aut}(S,C)|=1\} \text{ and } \\&{{\mathcal {N}}}{{\mathcal {T}}}(S)={\mathbb {P}}(S)\setminus {\mathcal {T}}(S). \end{aligned}$$

Since $bg (S)\subseteq \mathrm{Norm} _U(S)$ and ${\mathcal {T}}(S)\subseteq {\mathbb {P}}(S)$, we have

$$\begin{aligned} \mathrm{Norm} _U(S)\setminus bg (S)= & {} \{C\in {\mathbb {P}}(S)\mid U(S)_L\lhd \text {Aut}_C(S) \text{ and } \ |\text {Aut}(S,C)|\ne 1\}\\\subseteq & {} \{C\in {\mathbb {P}}(S)\mid \ |\text {Aut}(S,C)|\ne 1\}={\mathbb {P}}(S) \setminus {\mathcal {T}}(S)={{\mathcal {N}}}{{\mathcal {T}}}(S). \end{aligned}$$

Hence, for every monoid S we have

$$\begin{aligned} {|\mathrm{Norm} _U(S)\setminus bg (S)|\over 2^{|S|}}\le {|{\mathbb {P}}(S) \setminus {\mathcal {T}}(S)| \over 2^{|S|}}={|{{\mathcal {N}}}{{\mathcal {T}}}(S)|\over 2^{|S|}}. \end{aligned}$$

This implies that

$$\begin{aligned} {|\mathrm{Norm} _U(S)|\over 2^{|S|}}={|\mathrm{Norm} _U(S)\setminus bg (S)|\over 2^{|S|}}+{|bg (S)|\over 2^{|S|}}\le {|{{\mathcal {N}}}{{\mathcal {T}}}(S)|\over 2^{|S|}}+ {|bg (S)|\over 2^{|S|}}\ \ \ \ \ \ \ \ \ \ \ (I). \end{aligned}$$

Now note that since for every group G, $bg (G)\subseteq {\mathcal {T}}(G)$, we have the following weaker version of Babai and Godsil’s conjecture

$$\begin{aligned} \lim _{n\rightarrow \infty }\min _{G\in {\mathcal {G}}_n} {|{\mathcal {T}}(G)|\over 2^{|G|}}=1 \end{aligned}$$

(equivalently, almost all Cayley digraphs of groups have regular affine automorphism groups).

Clearly, the variant of the above version of Conjecture 1.1 for monoids is the following problem.

Problem 2.9

Is $\lim _{n\rightarrow \infty }\min _{S\in {\mathcal {M}}_n} {|{\mathcal {T}}(S)|\over 2^{|S|}}=1$?

Also there is another weaker version of Babai and Godsil’s conjecture. As we stated before, if Babai and Godsil’s conjecture is true, then Xu’s conjecture is true, too. So a weaker version of Babai and Godsil’s conjecture is the following:

“almost all Cayley digraphs of groups are normal if and only if almost all Cayley digraphs of groups are DRRs” (or equivalently, “Xu’s conjecture is equivalent to Babai and Godsil’s conjecture”).

Studying these weaker versions of Babai and Godsil’s conjecture for groups and monoids is one of our main goals. The paper is organized as follows. First, for a class ${\mathcal {F}}$ of monoids, we determine when

$$\begin{aligned} \lim _{n\rightarrow \infty }\min _{S\in {\mathcal {M}}_n\cap {\mathcal {F}}} {|{\mathcal {T}}(S)|\over 2^{|S|}}=1 \end{aligned}$$

(see Theorem 3.4). Then, we use ranks and large ranks of monoids to present some upper bounds of $\max _{S\in {\mathcal {M}}_n}{|{{\mathcal {N}}}{{\mathcal {T}}}(S)|\over 2^{|S|}}=1-\min _{S\in {\mathcal {M}}_n} {|{\mathcal {T}}(G)|\over 2^{|S|}}$ (see Theorem 2.11). To this end, as the first step, we use Remark 2.4 and we connect it to large ranks of monoids by proving the following result.

Theorem 2.10

For every $n\in {\mathbb {N}}$ and every class of monoids ${\mathcal {F}}$, we have

$$\begin{aligned} 0\le \max _{S\in {\mathcal {M}}_n\cap {\mathcal {F}}}{|{{\mathcal {N}}}{{\mathcal {T}}}(S)|\over 2^n}\le & {} {1\over 2^n} \max _{S\in {\mathcal {M}}_n\cap {\mathcal {F}}} (|{\mathcal {O}}(S)|\times \left( \max _{P\in {\mathcal {P}}(S)}\left( 2^{|\text {Fix}(P)|}\times 2^{{|S|-|\text {Fix}(P)|\over |P|}}\right) \right) \\\le & {} {1\over 2^{n\over 2}} \max _{S\in {\mathcal {M}}_n\cap {\mathcal {F}}} \left( |{\mathcal {O}}(S)|\times 2^{{r_5(S)\over 2}}\right) . \end{aligned}$$

In the next step, we connect $\max _{S\in {\mathcal {M}}_n}{|{{\mathcal {N}}}{{\mathcal {T}}}(S)|\over 2^{|S|}}$ to ranks of monoids and the cardinalities of the automorphism groups of monoids.

Theorem 2.11

For every class of monoids ${\mathcal {F}}$, we have

$$\begin{aligned} 0\le \max _{S\in {\mathcal {M}}_n\cap {\mathcal {F}}}{|{{\mathcal {N}}}{{\mathcal {T}}}(S)|\over 2^{|S|}}\le & {} {1\over 2^{{n\over 2}+1}} \max _{S\in {\mathcal {M}}_n \cap {\mathcal {F}}} \left( |\text {Aut}(S)| \times 2^{{r_5(S)\over 2}}\right) \\\le & {} {1\over 2^{{n\over 2}+1}} \max _{S\in {\mathcal {M}}_n\cap {\mathcal {F}}} \left( n^{\text {rank}(S)}\times 2^{{r_5(S)\over 2}}\right) . \end{aligned}$$

Using this theorem, we prove that Babai and Godsil’s conjecture is equivalent to Xu’s conjecture (see Theorem 1.7).

Then, we give a complete affirmative answer to Problem 1.5 (see Theorem 4.3). Also we use the results in the previous section, to illustrate the relation between normal and U-normal Cayley digraphs of monoids (see Theorem 4.8). Using this result, we explicitly determine the automorphism groups of vertex-transitive Cayley digraphs of monoids [(see Proposition 4.7]. Furthermore, by Theorem 4.3, we show that the answer to Problem 1.4 for Cayley digraphs of monoids is affirmative (see Proposition 4.6). Finally, in Remark 4.9, we state the origin of our approach which is based on the notion of closure operators.

3 Normal Cayley digraphs and DRRs

In this section for a monoid S, to present upper bounds of $|{{\mathcal {N}}}{{\mathcal {T}}}(S)|$, first we decompose ${{\mathcal {N}}}{{\mathcal {T}}}(S)$ into simpler subsets. Then, we continue our discussion by calculating the cardinalities of these subsets.

Remark 3.1

• (i) Let S be a monoid. Note that if $C^*$ is a minimal generating set of S, then none of non-identity automorphisms of S cannot fix all elements of $C^*$. Since the image of the elements of $C^*$ under every non-identity automorphism must be distinct elements in $S\setminus \{1\}$, we have $|\text {Aut}(S)|\le \prod _{i=1}^{|C^*|}(|S|-i)$. Therefore, by the definition of $\text {rank}(S)$, we have

  $$\begin{aligned} |\text {Aut}(S)|\le |S|^{\text {rank}(S)}. \end{aligned}$$

• (ii) For a group G, recall that $|\text {Aut}(G)|\le 2^{(\log _2(|G|))^2}$.

• (iii) For every group G, the number of minimal subgroups of G is less than or equal to ${|G|-1\over p-1}$, where p is the smallest prime divisor of |G|.

Lemma 3.2

Let S be a monoid and $P\in {\mathcal {P}}(S)$. Then, for every subset $C\subseteq S$ we have

$$\begin{aligned} U(S)_L\ne \text {Aut}_{P\cdot C}(S) \text{ and } P\le \text {Aut}_{P\cdot C}(S). \end{aligned}$$

Proof

Note that $P\cdot (P\cdot C)=P\cdot C$. Therefore, $\{\text {id}_S\}\ne P\le \text {Aut}_{P\cdot C}(S)$ and so $U(S)_L\lneqq \text {Aut}_{P\cdot C}(S)$ (note that $1\in \text {Fix}(P)$ and $1\not \in \text {Fix}(\lambda _u)$ for every $u\in U(S)$). $\square$

The next remark shows a special counting method which is our main tool for presenting an upper bound of ${|{{\mathcal {N}}}{{\mathcal {T}}}(S)|\over 2^{|S|}}$.

Remark 3.3

Recall that ${{\mathcal {N}}}{{\mathcal {T}}}(S)=\{C\subseteq S\mid |\text {Aut}(S,C)|\ne 1\}$. Equivalently, we can write

$$\begin{aligned} {{\mathcal {N}}}{{\mathcal {T}}}(S)= & {} \{C\subseteq S\mid \ \exists P\in {\mathcal {P}}(S) \text{ such } \text{ that } P\cdot C=C\}\\= & {} \cup _{P\in {\mathcal {P}}(S)} N(P). \end{aligned}$$

Therefore, to find the cardinality of ${{\mathcal {N}}}{{\mathcal {T}}}(S)$ we can use inclusion–exclusion principal to calculate it by using |N(P)|, where $P\in {\mathcal {P}}(S)$.

Now we present an equivalent condition for Problem 2.9. We will use the next result in proving Theorem  1.7.

Theorem 3.4

For every class ${\mathcal {F}}$ of finite monoids, the following conditions are equivalent.

• (i) $\lim _{n\rightarrow \infty }\min _{S\in {\mathcal {M}}_n\cap {\mathcal {F}}} {|bg (S)|\over 2^{|S|}}{=}1$ if and only if   $\lim _{n\rightarrow \infty }\min _{S\in {\mathcal {M}}_n\cap {\mathcal {F}}} {|\mathrm{Norm} _U(S)|\over 2^{|S|}}{=}1$;

• (ii)
  $$\begin{aligned} \lim _{n\rightarrow \infty }\min _{S\in {\mathcal {M}}_n\cap {\mathcal {F}}} {|{\mathcal {T}}(S)|\over 2^{|S|}}=1; \end{aligned}$$
• (iii)
  $$\begin{aligned} \lim _{n\rightarrow \infty }\max _{S\in {\mathcal {M}}_n\cap {\mathcal {F}}} {|{{\mathcal {N}}}{{\mathcal {T}}}(S)|\over 2^{|S|}}=0. \end{aligned}$$

Proof

By Remark 3.3 and the definitions of ${{\mathcal {N}}}{{\mathcal {T}}}(S)$ and ${\mathcal {T}}(S)$, conditions (ii) and (iii) are equivalent.

((i) $\Rightarrow$ (ii)) Note that if $\lim _{n\rightarrow \infty }\min _{S\in {\mathcal {M}}_n\cap {\mathcal {F}}} {|\mathrm{Norm} _U(S)|\over 2^{|S|}}=1$, then

$$\begin{aligned} \lim _{n\rightarrow \infty }\min _{S\in {\mathcal {M}}_n\cap {\mathcal {F}}} {|bg (S)|\over 2^{|S|}}=1. \end{aligned}$$

Now since $bg (S)\subseteq {\mathcal {T}}(S)$, we have $\lim _{n\rightarrow \infty } \min _{S\in {\mathcal {M}}_n\cap {\mathcal {F}}} {|{\mathcal {T}}(S)|\over 2^{|S|}}=1$.

((iii) $\Rightarrow$(i)) First, note that by inequality (I), we have

$$\begin{aligned} \min _{S\in {\mathcal {M}}_n\cap {\mathcal {F}}}{|\mathrm{Norm} _U(S)|\over 2^{|S|}}\le & {} \min _{S\in {\mathcal {M}}_n\cap {\mathcal {F}}}\left( {|{{\mathcal {N}}}{{\mathcal {T}}}(S)|\over 2^{|S|}}+ {|bg (S)|\over 2^{|S|}}\right) . \end{aligned}$$

Suppose that for $S_1$ and $S_2$ in ${\mathcal {M}}_n\cap {\mathcal {F}}$ we have

$$\begin{aligned} \left( {|{{\mathcal {N}}}{{\mathcal {T}}}(S_1)|\over 2^{|S_1|}}+ {|bg (S_1)|\over 2^{|S_1|}}\right) =\min _{S\in {\mathcal {M}}_n\cap {\mathcal {F}}} \left( {|{{\mathcal {N}}}{{\mathcal {T}}}(S)|\over 2^{|S|}}+ {|bg (S)|\over 2^{|S|}}\right) \end{aligned}$$

and

$$\begin{aligned} {|bg (S_2)|\over 2^{|S_2|}}=\min _{S\in {\mathcal {M}}_n\cap {\mathcal {F}}} {|bg (S)|\over 2^{|S|}}. \end{aligned}$$

So we have

$$\begin{aligned} \min _{S\in {\mathcal {M}}_n\cap {\mathcal {F}}}\left( {|{{\mathcal {N}}}{{\mathcal {T}}}(S)|\over 2^{|S|}}+ {|bg (S)|\over 2^{|S|}}\right)\le & {} \left( {|{{\mathcal {N}}}{{\mathcal {T}}}(S_2)|\over 2^{|S_2|}}+ {|bg (S_2)|\over 2^{|S_2|}}\right) \\\le & {} \max _{S\in {\mathcal {M}}_n\cap {\mathcal {F}}} {|{{\mathcal {N}}}{{\mathcal {T}}}(S)|\over 2^{|S|}}+ \min _{S\in {\mathcal {M}}_n\cap {\mathcal {F}}}{|bg (S)|\over 2^{|S|}}. \end{aligned}$$

Now for the non-trivial part of the proof, we just need to show that if

$$\begin{aligned} \lim _{n\rightarrow \infty } \min _{S\in {\mathcal {M}}_n\cap {\mathcal {F}}}{|\mathrm{Norm} _U(S)|\over 2^{|S|}}=1, \end{aligned}$$

then $\lim _{n\rightarrow \infty } \min _{S\in {\mathcal {M}}_n\cap {\mathcal {F}}}{|bg (S)|\over 2^{|S|}}=1$. Since by the above discussion

$$\begin{aligned} \min _{S\in {\mathcal {M}}_n\cap {\mathcal {F}}}{|\mathrm{Norm} _U(S)|\over 2^{|S|}}- \max _{S\in {\mathcal {M}}_n\cap {\mathcal {F}}}{|{{\mathcal {N}}}{{\mathcal {T}}}(S)|\over 2^{|S|}}\le \min _{S\in {\mathcal {M}}_n\cap {\mathcal {F}}}{|bg (S)|\over 2^{|S|}}, \end{aligned}$$

we have

$$\begin{aligned} 1=1-0= & {} \lim _{n\rightarrow \infty }\min _{S\in {\mathcal {M}}_n\cap {\mathcal {F}}}{|\mathrm{Norm} _U(S)|\over 2^{|S|}}- \lim _{n\rightarrow \infty }\max _{S\in {\mathcal {M}}_n \cap {\mathcal {F}}}{|{{\mathcal {N}}}{{\mathcal {T}}}(S)|\over 2^{|S|}}\\\le & {} \lim _{n\rightarrow \infty }\min _{S\in {\mathcal {M}}_n\cap {\mathcal {F}}}{|bg (S)|\over 2^{|S|}}\le 1\\ \end{aligned}$$

and the result is clear. $\square$

In the next result, for a monoid S we determine N(P), where $P\in {\mathcal {P}}(S)$.

Lemma 3.5

For a monoid S, let $P\in {\mathcal {P}}(S)$. Then

$$\begin{aligned} |N(P)|=2^{|\text {Fix}(P)|}\times 2^{{|S\setminus \text {Fix}(P)|\over |P|}}. \end{aligned}$$

Proof

Note that for each $y\in \text {Fix}(P)$, the subset $\{y\}$ belongs to N(P). Also if $z\in \text {NFix}(P)$, then $P\cdot z$ is the smallest element of N(P) which contains $\{z\}$. Obviously, if $z'\in P\cdot z$, then $P\cdot z$ is the smallest element of N(P) which contains $\{z'\}$. Note that since $z\not \in \text {Fix}(P)$, we have $|P\cdot z|=|P|$. Let $\{P\cdot g\mid g\in B\}$ be a partition of $\text {NFix}(P)$, where $B\subseteq S$. Clearly, $|B|=|\text {NFix}(P)|/|P|=|S\setminus \text {Fix}(P)|/|P|$. If $C'\in N(P)$, then $C'\cup (\cup _{b\in B'}P\cdot b)\in N(P)$, where $B'\subseteq B$. Also note that for every $C'\in N(P)$ and $X\subseteq \text {Fix}(P)$, the subset $C'\cup X$ belongs to N(P). Therefore, S is partitioned into $|\text {Fix}(P)|$ orbits of size one, and $|\text {NFix}(P)|/|P|$ orbits of size |P|. Since each element of |N(P)| is the union of some orbits of size 1 or |P|, we get that

$$\begin{aligned} | N(P)|= 2^{|\text {Fix}(P)|}\times 2^{|B|} =2^{|\text {Fix}(P)|} \times 2^{{|S\setminus \text {Fix}(P)|\over |P|}}. \end{aligned}$$

$\square$

Notation 3.6

For a monoid S, let

$$\begin{aligned} \nu (S)= \{N(P)\mid P\in {\mathcal {P}}(S)\}. \end{aligned}$$

Clearly, $|\nu (S)|\le |{\mathcal {P}}(S)|$. Also note that by the following example, the case $|\nu (S)|< |{\mathcal {P}}(S)|$, is possible.

Example 3.7

Let $G=C^3_2$ be the elementary abelian of order 8. Then $\text {Aut}(G)\cong \text {GL}(3,2)$ and every element of order 7 in $\text {Aut}(G)$ has a single orbit on the non-identity elements of G. Since $\text {GL}(3,2)$ has 8 subgroups of order 7, there exist two distinct automorphisms $\phi ,\psi \in \text {Aut}(G)$ of order 7 such that for every $g\in G$

$$\begin{aligned} \{\phi (g),\dots ,\phi ^7(g)\}=\{\psi (g),\dots ,\psi ^7(g)\} \end{aligned}$$

and the subgroups generated by $\phi$ and $\psi$ in $\text {Aut}(G)$ are distinct.

Note that having the cardinality of N(P) for every $P\in {\mathcal {P}}(S)$, we can use inclusion–exclusion principle and conclude the following result.

Lemma 3.8

For every monoid S, the cardinality of ${{\mathcal {N}}}{{\mathcal {T}}}(S)$ is equal to

$$\begin{aligned} \left( \sum _{N(P_i)\in \nu (S)} |N(P_i)|\right)- & {} \left( \sum _{N(P_i), N(P_{j'})\in \nu (S)} |N(\langle P_i\cup P_{i'}\rangle )|\right) \\+ & {} \cdots + (-1)^{|\nu (S)|} |N(\langle \cup _{P\in {\mathcal {P}}(S)}P\rangle )|. \end{aligned}$$

Proof

To prove the assertion, we use inclusion–exclusion principle. Let $P,P'\in {\mathcal {P}}(S)$. We show that $N(P)\cap N(P')=N(\langle P\cup P'\rangle )$. Obviously, $N(\langle P\cup P'\rangle )\subseteq N(P)\cap N(P')$. Let $C\subseteq S$ such that $P\cdot C=P'\cdot C=C$. Hence, $\langle P\cup P'\rangle \cdot C=C$ and specially, $C\in N(\langle P\cup P'\rangle )$. By induction, we get that

$$\begin{aligned} \cap _{i=1}^k N(P_i)=N(\langle \cup _{i=1}^k P_i\rangle ). \end{aligned}$$

$\square$

Note that by the proof of Lemma 3.5, it is clear that if there exist two distinct minimal subgroups $P,P'\in {\mathcal {P}}(S)$ such that $\beta (P)=\beta (P')$, then $N(P)=N(P')$ (see Example 3.7). So for calculating ${{\mathcal {N}}}{{\mathcal {T}}}(S)$ we must consider ${\mathcal {O}}(S)$, which is determined by ${\mathcal {P}}(S)$. Now by Lemma 3.8, we have the following result.

Corollary 3.9

For every monoid S, we have

$$\begin{aligned} |{{\mathcal {N}}}{{\mathcal {T}}}(S)|\le |{\mathcal {O}}(S)|\times \left( \max \left\{ 2^{|\text {Fix}(P)|}\times 2^{{|S|-|\text {Fix}(P)|\over |P|}}: P\in {\mathcal {P}}(S)\right\} \right) . \end{aligned}$$

Finally, we are in position to prove our main results in this section.

Proof of Theorem 2.10

By Corollary 3.9 we have

$$\begin{aligned} {|{{\mathcal {N}}}{{\mathcal {T}}}(S)|\over 2^{|S|}}\le {|{\mathcal {O}}(S)|\times \left( \max \{2^{|\text {Fix}(P)|}\times 2^{{|S|-|\text {Fix}(P)|\over |P|}}: P\in {\mathcal {P}}(S)\}\right) \over 2^{|S|}}. \end{aligned}$$

Now suppose that $P^*\in {\mathcal {P}}(S)$ such that

$$\begin{aligned} \max \{2^{|\text {Fix}(P)|}\times 2^{{|S|- |\text {Fix}(P)|\over |P|}}: P\in {\mathcal {P}}(S)\}= 2^{|\text {Fix}(P^*)|}\times 2^{{|S|-|\text {Fix}(P^*)|\over |P^*|}}. \end{aligned}$$

By Remark 2.4, we have $|S|-r_5(S)\le |S| -|\text {Fix}(P^*)|$. Therefore, we have

$$\begin{aligned} {|{{\mathcal {N}}}{{\mathcal {T}}}(S)|\over 2^{|S|}}\le & {} {|{\mathcal {O}}(S)|\times 2^{|\text {Fix}(P^*)|}\times 2^{{|S|-|\text {Fix}(P^*)|\over |P^*|}}\over 2^{|S|}}\\= & {} {|{\mathcal {O}}(S)|\times 2^{{|S|-|\text {Fix}(P^*)|\over |P^*|}}\over 2^{|S|-|\text {Fix}(P^*)|}}= {|{\mathcal {O}}(S)|\over 2^{(|S|-|\text {Fix}(P^*)|)\times {|P^*|-1 \over |P^*|}}}\\\le & {} {|{\mathcal {O}}(S)|\over 2^{(|S|-r_5(S))\times {|P^*|-1 \over |P^*|}}}. \end{aligned}$$

Since $|P^*|\ge 2$, we have

$$\begin{aligned} {|{{\mathcal {N}}}{{\mathcal {T}}}(S)|\over 2^{|S|}}\le {|{\mathcal {O}}(S)|\over 2^{|S|-r_5(S) \over 2}}. \end{aligned}$$

Therefore, for every $n\in {\mathbb {N}}$ we have

$$\begin{aligned} \max _{S\in {\mathcal {M}}_n\cap {\mathcal {F}}}{|{{\mathcal {N}}}{{\mathcal {T}}}(S)|\over 2^{|S|}}\le \max _{S\in {\mathcal {M}}_n \cap {\mathcal {F}}}{|{\mathcal {O}}(S)|\over 2^{|S|-r_5(S) \over 2}}= {1\over 2^{n\over 2}} \max _{S\in {\mathcal {M}}_n\cap {\mathcal {F}}} \left( |{\mathcal {O}}(S)|\times 2^{{r_5(S)\over 2}}\right) . \end{aligned}$$

$\square$

Now using Theorem 2.10, we prove the next result.

Proof of Theorem 2.11

Note that by Remark 3.1(i), for every monoid S, we have $|\text {Aut}(S)|\le |S|^{\text {rank}(S)}$. On the other hand, by Remark 3.1(ii), we can conclude that $|{\mathcal {O}}(S)|\le |{\mathcal {P}}(S)|\le {|\text {Aut}(S)|/ 2}\le |S|^{\text {rank}(S)}/2$. Now by Theorem 2.10 we have

$$\begin{aligned} 0\le \max _{S\in {\mathcal {M}}_n\cap {\mathcal {F}}}{|{\mathcal {N}}(S)|\over 2^{|S|}}\le & {} {1\over 2^{{n\over 2}}} \max _{S\in {\mathcal {M}}_n\cap {\mathcal {F}}} \left( |{\mathcal {O}}(S)|\times 2^{{r_5(S)\over 2}}\right) \\\le & {} {1\over 2^{{n\over 2}+1}} \max _{S\in {\mathcal {M}}_n\cap {\mathcal {F}}} \left( |\text {Aut}(S)|\times 2^{{r_5(S)\over 2}}\right) \\\le & {} {1\over 2^{{n\over 2}+1}} \max _{S\in {\mathcal {M}}_n\cap {\mathcal {F}}} \left( n^{\text {rank}(S)}\times 2^{{r_5(S)\over 2}}\right) . \end{aligned}$$

$\square$

Now by considering the class $FGrp$, as a consequence of Theorem 2.11, we prove that Xu’s conjecture is equivalent to Babai and Godsil’s conjecture.

Proof of Theorem 1.7

By Remark 3.1(ii), for every group G we have $|\text {Aut}(G)|\le 2^{(\log _2(|G|))^2}$. On the other hand, note that by Theorem 1.6, $r_5(G)\le {|G|\over 2}+1$. Since for every $n\in {\mathbb {N}}$, we have ${\mathcal {M}}_n\cap FGrp ={\mathcal {G}}_n$, by Theorem 2.11, we have

$$\begin{aligned} 0\le \lim _{n\rightarrow \infty }\max _{S\in {\mathcal {M}}_n\cap FGrp }{|{{\mathcal {N}}}{{\mathcal {T}}}(S)|\over 2^{|S|}}\le & {} \lim _{n\rightarrow \infty }{1\over 2^{{n\over 2}+1}} \max _{G\in {\mathcal {G}}_n} \left( |\text {Aut}(G)|\times 2^{{r_5(G)\over 2}}\right) \\\le & {} \lim _{n\rightarrow \infty }{1\over 2^{{n\over 2}+1}} \max _{G\in {\mathcal {G}}_n} \left( 2^{(\log _2(|G|))^2}\times 2^{{{|G|\over 2}+1\over 2}}\right) \\\le & {} \lim _{n\rightarrow \infty }{1\over 2^{{n\over 2}+1}} \max _{G\in {\mathcal {G}}_n} \left( |G|^2\times 2^{{{|G|\over 2}+1\over 2}}\right) =0. \end{aligned}$$

So by Theorem 3.4, Xu’s conjecture is equivalent to Babai and Godsil’s conjecture.

$\square$

Note that by Theorem 1.7, the answer to Problem 2.9 is affirmative for the class ${{\mathcal {B}}}{{\mathcal {S}}}(1)= FGrp$. In the following, we present some other classes of monoids ${\mathcal {F}}$ such that $\lim _{n\rightarrow \infty }\min _{S\in {\mathcal {M}}_n\cap {\mathcal {F}}}{|{\mathcal {T}}(S)|\over 2^{|S|}}=1.$

Proposition 3.10

For every $m\in {\mathbb {N}}\setminus \{1\}$ and the family ${{\mathcal {B}}}{{\mathcal {S}}}^1(m)$, we have

$$\begin{aligned} \lim _{n\rightarrow \infty }\min _{S\in {\mathcal {M}}_n\cap {{\mathcal {B}}}{{\mathcal {S}}}^1(m)}{|{\mathcal {T}}(S)|\over 2^{|S|}}=1. \end{aligned}$$

Proof

Let $m\in {\mathbb {N}}\setminus \{1\}$. First, note that every $S\in {{\mathcal {B}}}{{\mathcal {S}}}(m)$ is not a monoid and $S^1=S\cup \{1\}\in {{\mathcal {B}}}{{\mathcal {S}}}^1(m)$. To prove the assertion, we use Theorem 3.4 and we show that

$$\begin{aligned} \lim _{n\rightarrow \infty }\max _{S\in {\mathcal {M}}_n\cap {{\mathcal {B}}}{{\mathcal {S}}}^1(m)}{|{{\mathcal {N}}}{{\mathcal {T}}}(S)|\over 2^{|S|}}=0. \end{aligned}$$

Let G be a group. Note that for $T=B(G,m)\in {{\mathcal {B}}}{{\mathcal {S}}}(m)$ and $S=T^1$, we have $\sigma \in \text {Aut}(S^1)$ if and only if $\sigma |_T\in \text {Aut}(T)$ and $\sigma (1)=1$. Also note that a subset $C\subseteq S=T^1$ is a generating set of S if and only if $1\in C$ and $C\setminus \{1\}$ is a generating set of T. Therefore, $r_5(S)=r_5(T)+1$ and $|\text {Aut}(S)|=|\text {Aut}(T)|$. Recall that by [31, Theorem 3.1] since $m\ge 2$, we know that $r_5(T)=r_5(B(G, m)) = (m^2- m + 1)|G| + 2$. On the other hand, by [7, Theorem 2] we have $|\text {Aut}(B(G,m))|= (m!)\times |\text {Aut}(G)|\times |G|^m$. Since $n=m^2|G|+1$, by the above discussions, Theorem 2.11 and Remark 3.1, the limit $\lim _{n\rightarrow \infty }\max _{S\in {\mathcal {M}}_n\cap {{\mathcal {B}}}{{\mathcal {S}}}^1(m)}{|{{\mathcal {N}}}{{\mathcal {T}}}(S)|\over 2^{|S|}}$ is less than or equal to

$$\begin{aligned}&\lim _{n\rightarrow \infty }{1\over 2^{{n\over 2}+1}} \max _{S\in {\mathcal {M}}_n\cap {{\mathcal {B}}}{{\mathcal {S}}}^1(m)} \left( m!\times |\text {Aut}(G)|\times |G|^m\times 2^{{(m^2- m + 1)|G| + 3\over 2}}\right) \\&\quad \le \lim _{n\rightarrow \infty } \max _{S\in {\mathcal {M}}_n\cap {{\mathcal {B}}}{{\mathcal {S}}}^1(m)} \left( {m!\times 2^{(\log _2(|G|))^2}\times |G|^m\times 2^{{(m^2- m + 1)|G| + 3\over 2}}\over 2^{{m^2|G|+1\over 2}+1}}\right) \\&\quad \le \lim _{n\rightarrow \infty } \max _{S\in {\mathcal {M}}_n\cap {{\mathcal {B}}}{{\mathcal {S}}}^1(m)} \left( {m!\times |G|^{m+2} \times 2^{{(m^2- m + 1)|G| \over 2}}\over 2^{m^2|G|\over 2} }\right) \\&\quad \le \lim _{n\rightarrow \infty } \max _{S\in {\mathcal {M}}_n\cap {{\mathcal {B}}}{{\mathcal {S}}}^1(m)} \left( {{m!\times |G|^{m+2}}\over 2^{{( m- 1)|G| \over 2}} }\right) =0\\ \end{aligned}$$

and the result is clear. $\square$

Now we present another family of monoids such that the answer to Problem 2.9 is affirmative. For this purpose, we recall some facts about monoids.

Let Y be a finite chain (a finite totally ordered set). The semigroup $(Y,\min )$ is a commutative and idempotent monoid with identity element 1 and zero element 0, where 1 and 0 are the maximum and minimum element of Y, respectively. From now on, for simplicity, by $Y_{\min }$ we mean the above monoid structure of a chain Y. Also let

$$\begin{aligned} Ch =\{Y_{\min }\mid Y \text{ is } \text{ a } \text{ chain }\}. \end{aligned}$$

We explain one of our main reasons for studying this class of monoids in Remark 4.5.

We want to show that

$$\begin{aligned} \lim _{n\rightarrow \infty }\min _{S\in {\mathcal {M}}_n\cap ( FGrp \times Ch )} {|{\mathcal {T}}(S)|\over 2^{|S|}}=1. \end{aligned}$$

For this purpose, first we calculate $|\text {Aut}( G\times Y_{\min })|$ for a group G and a chain Y.

Lemma 3.11

For a chain Y and a group G, the automorphism group of the monoid $G\times Y_{\min }$ is isomorphic to the product of |Y| copies of $\text {Aut}(G)$. In particular, $|\text {Aut}( G\times Y_{\min })|=|\text {Aut}(G)|^{|Y|}$.

Proof

Suppose that $\pi _1:G\times Y_{\min }\rightarrow G$ denotes the natural projection onto the first component. Now let $\sigma \in \text {Aut}(G\times Y_{\min })$ and $(g,y)\in G\times Y$. We prove the assertion in the following steps.

• Step (i)   We show that for every $y\in Y$, there exists $y'\in Y$ such that $\sigma (G\times \{y\})=G\times \{y'\}$.

• Step (ii)   We show that if $y_1\le y_2$ and $\sigma (G\times \{y_i\})=G\times \{y'_i\}$ for some $y'_i\in Y$, where $i=1,2$, then $y'_1\le y'_2$.

• Step (iii)    The function $\sigma _y:G\rightarrow G$ defined by $\sigma _y(g)=\pi _1(\sigma (g,y))$ is a group automorphism.

• Step (iv)    The function $\Theta :\text {Aut}_C(S)\rightarrow \prod _{y\in Y}\text {Aut}(G)$ defined by $\Theta (\sigma )=(\sigma _y)_{y\in Y}$, for every $\sigma \in \text {Aut}_C(S)$, is a group isomorphism.

(i) Suppose that $\sigma (g,y)=(g',y')$. Since there exists $m\in {\mathbb {N}}$ such that $g^m=1$, we have $\sigma (1_G,y)=\sigma (g^m,y)=\sigma (g^m,y^m)=(\sigma (g,y))^m=(g'^m,y'^m)=(g'^m,y')$. Since there exists $m'\in {\mathbb {N}}$ such that $g'^{m'}=1_G$, we have

$$\begin{aligned} (\sigma (g^m,y^m))^{m'}=((g'^m)^{m'},(y'^m)^{m'})=(1_G,y'). \end{aligned}$$

Hence, $\sigma (G\times \{y\})=G\times \{y'\}$.

(ii) Now note that since $\sigma$ is a monoid automorphism, we have $\sigma (1_G,1_Y)=(1_G,1_Y)$. On the other hand, suppose that for $y_1\le y_2$, we have $\sigma (1_G,y_i)=(1_G, y'_i)$ for $i=1,2$. Then, we have

$$\begin{aligned} (1_G,y'_1)= & {} \sigma (1_G,y_1)=\sigma (1_G,\min \{y_1,y_2\}) =\sigma ((1_G,y_1)(1_G,y_2))\\= & {} \sigma (1_G,y_1)\sigma (1_G,y_2)\\= & {} (1_G,y'_1)(1_G,y'_2)=(1_G,\min \{y'_1,y'_2\}).\\ \end{aligned}$$

Hence, $y'_1\le y'_2$ and this implies that $\sigma (1_G,y)=(1_G,y)$, because $\sigma$ is one-one and Y is a finite chain. So for every $y\in Y$, we have $\sigma (G\times \{y\})=G\times \{y\}$ and $\sigma (g,y)=(\pi _1(\sigma (g,y)),y)$ for every $g\in G$.

(iii) For $g,g'\in G$ and $y\in Y$, suppose that $(g'',y)=\sigma (g_1g_2,y)$ and $(g_i',y)=\sigma (g_i,y)$, where $g_i'\in G$ and $i=1,2$. Now we have

$$\begin{aligned} (g'',y)=\sigma (g_1g_2,y)=\sigma (g_1,y)\sigma (g_2,y)=(g_1',y)(g_2',y)=(g_1'g_2',y). \end{aligned}$$

Hence, the function $\sigma _y:G\rightarrow G$ defined by

$$\begin{aligned} \sigma _y(g)=\pi _1(\sigma (g,y)), \end{aligned}$$

is a group automorphism.

(iv) By Step (iii), the function $\Theta :\text {Aut}(G\times Y_{\min })\rightarrow \prod _{y\in Y}\text {Aut}(G)$ defined by $\Theta (\sigma )=(\sigma _y)_{y\in Y}$ is well-defined. Also since for every $\sigma ,\sigma '\in \text {Aut}(S)$, $g\in G$ and $y\in Y$ we have

$$\begin{aligned} (\sigma \sigma ')(g,y)=\sigma (\sigma '(g,y)) =\sigma (\sigma '_y(g),y)=(\sigma _y(\sigma '_y(g)),y)=(\sigma _y\sigma '_y(g),y), \end{aligned}$$

it is clear that $(\sigma \sigma ')_y=\sigma _y\sigma '_y$. Therefore, $\Theta$ is a group homomorphism. Finally, note that for every $(f_y)_{y\in Y}\in \prod _{y\in Y}\text {Aut}(G)$, the function $\sigma :G\times Y\rightarrow G\times Y$ defined by

$$\begin{aligned} \sigma (g,y)=(f_y(g),y). \end{aligned}$$

is a monoid automorphism. So $\Theta$ is onto and one-one. Hence, $\Theta$ is a group isomorphism.

Now the rest of the proof is clear. $\square$

An immediate consequence of the above lemma and Remark 3.1(ii), we have the following result.

Corollary 3.12

Let Y be a chain and G be a group. Then

$$\begin{aligned} |\text {Aut}(G\times Y_{\min })|\le \left( 2^{(\log _2(|G|))^2}\right) ^{|Y|}=|G|^{2|Y|}. \end{aligned}$$

Now we are able to give another affirmative answer to Problem 2.9. We will use this result to illustrate the relation between normal Cayley digraphs and U-normal Cayley digraphs.

Theorem 3.13

Let ${\mathcal {F}}= FGrp \times Ch$. Then

$$\begin{aligned} \lim _{n\rightarrow \infty }\min _{S\in {\mathcal {M}}_n\cap {\mathcal {F}}} {|{\mathcal {T}}(S)|\over 2^{|S|}}=1. \end{aligned}$$

Proof

We prove the assertion by Theorem 3.4. Let G be a group and Y be a chain such that $|Y|>1$. Note that $(1_G,1_Y)$ is the identity element of $S=G\times Y_{\min }$. Also note that for every $\sigma \in \text {Aut}(S)$, following the terminology in the proof of Lemma 3.11 we have

$$\begin{aligned} \text {Fix}(\sigma )=\{(g,y)\in S\mid \sigma (g,y)=(\sigma _y(g),y)=(g,y)\}=\cup _{y\in Y} (\text {Fix}(\sigma _y)\times \{y\}). \end{aligned}$$

So $|\text {Fix}(\sigma )|\le |Y|\times {|G|\over 2}$. Since

$$\begin{aligned} 2^{|\text {Fix}(\sigma )|}\times 2^{{|\text {NFix}(\sigma )|\over 2}}\le 2^{{|Y|\times |G|\over 2}}\times 2^{{|Y|\times {|G|\over 2}\over 2}}, \end{aligned}$$

by Corollary 3.12 we have

$$\begin{aligned} 0\le & {} \max _{G\times Y\in {\mathcal {M}}_n\cap {\mathcal {F}}}{|{{\mathcal {N}}}{{\mathcal {T}}}(G\times Y)|\over 2^n}\le {1\over 2^n} \max _{G\times Y\in {\mathcal {M}}_n\cap {\mathcal {F}}} \left( |{\mathcal {O}}(G\times Y)|\times 2^{{|Y|\times |G|\over 2}}\times 2^{{|Y|\times |G|\over 4}}\right) \\\le & {} {1\over 2^{n+1}} \max _{G\times Y\in {\mathcal {M}}_n\cap {\mathcal {F}}} \left( |\text {Aut}(G\times Y)|\times 2^{{|Y|\times |G|\over 2}}\times 2^{{|Y|\times |G|\over 4}}\right) .\\ \end{aligned}$$

By Lemma 3.11 since $n=|G|\times |Y|$, we have

$$\begin{aligned} 0\le & {} \max _{G\times Y\in {\mathcal {M}}_n\cap {\mathcal {F}}}{|{{\mathcal {N}}}{{\mathcal {T}}}(G\times Y)|\over 2^n} \le {1\over 2^{n+1}} \max _{G\times Y\in {\mathcal {M}}_n\cap {\mathcal {F}}} \left( |\text {Aut}(G)|^{|Y|}\times 2^{{|Y|\times |G|\over 2}}\times 2^{{|Y|\times |G|\over 4}}\right) \\\le & {} {1\over 2^{n+1}} \max _{G\times Y\in {\mathcal {M}}_n\cap {\mathcal {F}}} \left( |G|^{2|Y|}\times 2^{{|Y|\times |G|\over 2}}\times 2^{{|Y|\times |G|\over 4}}\right) \\\le & {} {1\over 2} \max _{G\times Y\in {\mathcal {M}}_n\cap {\mathcal {F}}} {\left( |G|^{2|Y|}\right) \over 2^{{|G|\times |Y|}\over 4}}= {1\over 2} \max _{G\times Y\in {\mathcal {M}}_n\cap {\mathcal {F}}} \left( {|G|^2\over 2^{{|G|}\over 4}}\right) ^{|Y|}. \end{aligned}$$

Clearly, there exists a natural number N such that for every group G such that $|G|\ge N$, we have

$$\begin{aligned} \left( {|G|^2\over 2^{{|G|}\over 4}}\right) <{1\over 2}. \end{aligned}$$

Since $|Y|>1$, for every group G such that $|G|\ge N$, we have

$$\begin{aligned} \left( {|G|^2\over 2^{{|G|}\over 4}}\right) ^{|Y|}\le ({1\over 2})^{|Y|}. \end{aligned}$$

On the other hand, we have

$$\begin{aligned} \left( {|G|^2\over 2^{{|G|}\over 4}}\right) ^{|Y|}\le {|G|^2\over 2^{{|G|}\over 4}}. \end{aligned}$$

Now note that if $n=|G|\times |Y|$ tends to infinity, then $|G|\rightarrow \infty$ or $|Y|\rightarrow \infty$. In either of these cases, by the above discussion we have

$$\begin{aligned} \lim _{n\rightarrow \infty }{1\over 2} \max _{G\times Y_{\min }\in {\mathcal {M}}_n\cap {\mathcal {F}}} \left( {|G|^2\over 2^{{|G|}\over 4}}\right) ^{|Y|} \le \lim _{n\rightarrow \infty }{1\over 2} \max _{G\times Y_{\min }\in {\mathcal {M}}_n\cap {\mathcal {F}}} \left( \min \left\{ \left( {1\over 2}\right) ^{|Y|},{|G|^2\over 2^{|G|\over 4}}\right\} \right. \end{aligned}$$

and so the result is clear. $\square$

4 The automorphism groups of vertex-transitive Cayley digraphs of monoids

In this section, we give affirmative answer to Problem 1.5 and we use it to answer Problem 1.4, and to determine automorphism groups of Cayley digraphs of monoids. Also we discuss the relation between normal and U-normal Cayley digraphs of monoids with respect to generating sets. Note that the known vertex-transitive Cayley digraphs of monoids are group digraphs (see, for example, [19, 20] or [25]). As the main results of this section, we show that every Cayley digraphs of monoids are group digraphs.

Before we start our investigation, we remark that although our results in this section are about Cayley digraphs of monoids, note that there are vertex-transitive Cayley digraphs of semigroups which are not monoids.

Example 4.1

Let S be a right-zero semigroup such that $|S|=n\ge 1$. The Cayley digraph $\text {Cay}(S,S)$ is isomorphic to $\overrightarrow{K_n}$, where $\overrightarrow{K_n}$ denotes the complete graph on n vertices with a loop incident to each of its vertices. Clearly, $\text {Cay}(S,S)$ satisfies the following conditions:

• (i) $\text {Cay}(S,S)$ is vertex-transitive;

• (ii) Its underlying undirected graph of $\text {Cay}(S,S)$ is a group graph;

• (iii) S is not a monoid (group).

Now we start the study of automorphism groups of vertex-transitive Cayley digraphs of monoids. As the first step, we characterize these Cayley digraphs by proving Theorem 4.3.

Notation 4.2

Let S be a semigroup, and let C be a subset of S. For every $s\in S$, by $\Gamma _s$ we mean the connected component of $\Gamma =\text {Cay}(S,C)$ which contains s. Also by $V(\Gamma _s)$, we mean the set of vertices of $\Gamma _s$. Finally, note that by $\mathrm{Gen} (S)$ we mean the collection of all generating sets of S.

In the next theorem, we give affirmative answer to Problem 1.5.

Theorem 4.3

Let S be a monoid and $C\subseteq S$. The Cayley digraph $\Gamma =\text {Cay}(S,C)$ is vertex-transitive if and only if

• (i) $\langle C\rangle$ is a subgroup of S and $1_{\langle C\rangle }=1_S$;

• (ii) every connected component of $\text {Cay}(S,C)$ is strongly connected;

• (iii) the number of connected components of $\text {Cay}(S,C)$ is equal to $|S|/|\langle C\rangle |$.

Proof

($\Rightarrow$) Suppose that $\text {Cay}(S,C)$ is vertex-transitive. We show that $V(\Gamma _1)=\langle C\rangle$. For every $s\in S$, let

$$\begin{aligned} D_s=\{t\in S\mid \exists c_1,\dots , c_n\in C \text{ such } \text{ that } t=sc_1\cdots c_n\}. \end{aligned}$$

Suppose that $s^*\in \langle C\rangle$ such that $|D_{s^*}|$ is the maximum of $\{|D_s|\mid s\in \langle C\rangle \}$. For every $x\in D_{s^*}$, we have $D_x\subseteq D_{s^*}$. Since $\text {Cay}(S,C)$ is vertex-transitive, there exists $\sigma \in \text {Aut}_C(S)$ such that $\sigma (s^*)=x$. Therefore, for every $x\in D_{s^*}$, we have $|D_x|\ge |D_{s^*}|$ and specially $D_x=D_{s^*}$. So the subdigraph induced by $D_{s^*}$ in $\Gamma$ is strongly connected. If $V(\Gamma _1)=V(\Gamma _{s^*})=D_{s^*}$, then we have nothing to prove. Otherwise, suppose that $y\in V(\Gamma _1)\setminus D_{s^*}$ and y is an in-neighbor of some $x\in D_{s^*}$. Since $D_{s^*}\subseteq D_y$, by the maximality of $|D_{s^*}|$ we have $D_{s^*}=D_y$, which is a contradiction. Hence, $V(\Gamma _1)=D_{s^*}=D_1=\langle C\rangle$. On the other hand, by the above discussion, since for every $c\in C$ we have $D_c=D_1$, every element of C is right invertible. Hence, $\langle C\rangle$ is a group and $1_{\langle C\rangle }=1_S$.

For every $s\in S$, since $\Gamma$ is vertex-transitive, the connected component $\Gamma _s$ is isomorphic to $\Gamma _1$. Therefore, every connected component of $\Gamma$ is strongly connected. Finally, note that the number of connected components of $\Gamma$ is equal to $|S|/|\langle C\rangle |$, because $|V(\Gamma _s)|=|\langle C\rangle |$ for every $s\in S$.

($\Leftarrow )$ By (i), $\text {Cay}(\langle C\rangle , C)$ is a group digraph and so it is vertex-transitive. For every $s\in S$ note that by (ii), the connected component $\Gamma _s$ is strongly connected and therefore we have $V(\Gamma _s)=s\langle C\rangle$. So the function $f_s:\langle C\rangle \rightarrow V(\Gamma _s)$ defined by $f_s(c_1\cdots c_n)=sc_1\cdots c_n$ is onto. Hence, $|V(\Gamma _s)|\le |\langle C\rangle |$. Since by (iii), the number of connected components of $\Gamma$ is equal to $|S|/|\langle C\rangle |$, we have $|V(\Gamma _s)|=|\langle C\rangle |$. Therefore, $f_s:\langle C\rangle \rightarrow V(\Gamma _s)$ is one-one. Since $f_s:\Gamma _1\rightarrow \Gamma _s$ is clearly a digraph homomorphism, by the above discussion it is a digraph isomorphism. Therefore, $\Gamma _s$ is isomorphic to $\Gamma _1$ and it is a group digraph, too. Note that every disjoint union of mutually isomorphic group digraphs, is again a group digraph. So the result is clear. $\square$

As a consequence of the above theorem, we can generalize [19, Theorem 3.7] and [20, Theorem 3.4].

Proposition 4.4

Let S be a monoid and $C\subseteq S$. Let $\Gamma =\text {Cay}(S,C)$. The following conditions are equivalent.

• (i) $\text {Cay}(S,C)$ is $\text {ColAut}_C(S)$-vertex-transitive;

• (ii) $\text {Cay}(S,C)$ is $\text {CPAut}_C(S)$-vertex-transitive;

• (iii) $\text {Cay}(S,C)$ is vertex-transitive

• (iv) $\text {Cay}(\langle C\rangle ,C)$ is a group digraph and for every $s\in S$, the connected component $\Gamma _s$ is isomorphic to $\Gamma _1$.

Remark 4.5

As an immediate consequence of Theorem 4.3, it is easy to see that the family of vertex-transitive Cayley digraphs of monoids is equal to the family of group digraphs. On the other hand, it is equal to the family of vertex-transitive Cayley digraphs of monoids in $FGrp \times Ch$ (note that for every $S=G\times Y_{\min }\in FGrp \times Ch$, we have $U(G\times Y_{\min })=G\times \{1\}$). So by Theorem 4.3, if for some subset C of S, the Cayley digraph $\text {Cay}(S,C)$ is vertex-transitive, then $C\subseteq G\times \{1\}$. Conversely, for every $C\subseteq U(S)$, the Cayley digraph $\text {Cay}(S,C)$ is vertex-transitive). This is one of our reasons for studying this class of monoids.

Now by Theorem 4.3, we can solve Problem 1.4.

Proposition 4.6

Let S be a monoid and $C\subseteq S$. The following conditions are equivalent.

• (i) $\text {Cay}(S,\{c\})$ is vertex-transitive, for every $c\in C$;

• (ii) $\text {Cay}(S,C)$ is $\text {ColAut}_C(S)$-vertex-transitive.

Proof

For the non-trivial part of the proof, we show that if $\text {Cay}(S,\{c\})$ is vertex-transitive for every $c\in C$, then $\text {Cay}(S,C)$ is $\text {ColAut}_C(S)$-vertex-transitive. We prove the assertion by Theorem 4.3. For this purpose, note that by Theorem 4.3, since for every $c\in C$, the Cayley digraph $\text {Cay}(S,\{c\})$ is vertex-transitive, the element c belongs to U(S). So $\langle C\rangle \subseteq U(S)$. On the other hand, since $\langle C\rangle$ is a subgroup of S and $1_S=1_{\langle C\rangle }$, every connected component of $\text {Cay}(S,\{c\})$ is strongly connected. Finally, we show that for every $s\in S$, the connected component $\Gamma _s$ is isomorphic to $\Gamma _1=\text {Cay}(\langle C\rangle , C)$. First, we show that for every $c,c'\in C$ if $sc=sc'$, then $c=c'$. Define $\phi _s:\Gamma _1\rightarrow \Gamma _s$ as follows $\phi _s(1)=s$ and for every $c_1,\dots ,c_n\in C$, let $\phi _s(c_1\cdots c_n)=sc_1\cdots c_n$. By the above discussion, $\phi _s$ is a digraph isomorphism. Let $\psi _s:\Gamma \rightarrow \Gamma$ be defined by

$$\begin{aligned} \psi _s(x) = {\left\{ \begin{array}{ll} {\phi _s(x),} &{}\quad \text {if } x\in \langle C\rangle ;\\ {\phi _s^{-1}(x),} &{}\quad \text {if } x\in s\langle C\rangle ;\\ \text {x,} &{}\quad \text {otherwise.} \end{array}\right. } \end{aligned}$$

Clearly, $\psi _s\in \text {ColAut}_C(S)$ and $\psi _s(1)=s$. Therefore, $\text {Cay}(S,C)$ is $\text {ColAut}_C(S)$-vertex-transitive. $\square$

Now we use Theorem 4.3 to characterize the automorphism groups of vertex-transitive Cayley digraphs of monoids. Before we continue our discussion, we recall the following notions which are needed in the sequel. For every $n\in {\mathbb {N}}$, the symmetric group on a finite set of n symbols is denoted by $S_n$. Let K and H be groups. Let $P=\prod _{h\in H} K_h$ of copies of $K_h=K$ indexed by H. Define $k_hk'_{h'}=(kk')_{hh'}$ for every $k_h\in K_h$ and $k'_{h'}\in K_{h'}$. This group is called the wreath product of K and H and is denoted by $K \text { wr } H$.

Now by the definition of the wreath product of two groups and the above theorem, we have the following result.

Proposition 4.7

Let S be a monoid and $C\subseteq S$. Suppose that $\text {Cay}(S,C)$ is vertex-transitive. Then, the automorphism group of $\text {Cay}(S,C)$ is isomorphic to $\text {Aut}_C(\langle C\rangle ) \text { wr } S_d$, where $d=|S|/|\langle C\rangle |$.

Now we use the results in the previous section, to establish relations between normal and U-normal Cayley digraphs. Also this result shows one of the values of studying Problem 2.9.

Theorem 4.8

Let ${\mathcal {F}}= FGrp \times Ch$. The following conditions are equivalent.

• (i) Almost all Cayley digraphs of monoids in ${\mathcal {F}}$ with respect to generating sets are normal, i.e.,

  $$\begin{aligned} \lim _{n\rightarrow \infty }\min _{S\in {\mathcal {M}}_n\cap {\mathcal {F}}} {|\mathrm{Norm} (S)\cap \mathrm{Gen} (S)|\over | \mathrm{Gen} (S)|}=1; \end{aligned}$$

• (ii) Almost all Cayley digraphs of monoids in ${\mathcal {F}}$ with respect to generating sets are U-normal, i.e.,

  $$\begin{aligned} \lim _{n\rightarrow \infty }\min _{S\in {\mathcal {M}}_n\cap {\mathcal {F}}} {|\mathrm{Norm} _U(S)\cap \mathrm{Gen} (S)|\over | \mathrm{Gen} (S)|}=1; \end{aligned}$$
• (iii)
  $$\begin{aligned} \lim _{n\rightarrow \infty }\min _{S\in {\mathcal {M}}_n\cap {\mathcal {F}}} {|bg (S)\cap \mathrm{Gen} (S)|\over | \mathrm{Gen} (S)|}=1; \end{aligned}$$

Proof

By Lemma 2.5, conditions (i) and (ii) are equivalent.

((ii) $\Rightarrow$ (iii)) For a monoid S, note that

$$\begin{aligned} \mathrm{Norm} _U(S)\cap \mathrm{Gen} (S)= & {} (bg (S)\cap \mathrm{Gen} (S))\cup ((\mathrm{Norm} _U(S)\setminus bg (S))\cap \mathrm{Gen} (S))\\\subseteq & {} (bg (S)\cap \mathrm{Gen} (S))\cup ({{\mathcal {N}}}{{\mathcal {T}}}(S)\cap \mathrm{Gen} (S)) \end{aligned}$$

On the other hand, since for every generating set C of S, there exists at least one minimal generating set $C^*\subseteq C$, we have

$$\begin{aligned} | \mathrm{Gen} (S)|\ge 2^{|S|-\text {rank}(S)}=|\{C'\subseteq S\mid C_0^*\subseteq C'\}| \end{aligned}$$

in which $C_0^*$ is a generating set of S such that $|C_0^*|=\text {rank}(S)$. Therefore, we have

$$\begin{aligned} \min _{S\in {\mathcal {M}}_n\cap {\mathcal {F}}}\Bigg ({|\mathrm{Norm} _U(S)\cap \mathrm{Gen} (S)|\over | \mathrm{Gen} (S)|}\Bigg )\le & {} \min _{S\in {\mathcal {M}}_n\cap {\mathcal {F}}}{|bg (S)\cap \mathrm{Gen} (S)|\over | \mathrm{Gen} (S)|}\\&+ \max _{S\in {\mathcal {M}}_n\cap {\mathcal {F}}}{|{{\mathcal {N}}}{{\mathcal {T}}}(S)\cap \mathrm{Gen} (S)|\over | \mathrm{Gen} (S)|}\\\le & {} \min _{S\in {\mathcal {M}}_n\cap {\mathcal {F}}}{|bg (S)\cap \mathrm{Gen} (S)|\over | \mathrm{Gen} (S)|}\\&+ \max _{S\in {\mathcal {M}}_n\cap {\mathcal {F}}}{|{{\mathcal {N}}}{{\mathcal {T}}}(S)\cap \mathrm{Gen} (S)|\over 2^{|S|-\text {rank}(S)}}. \end{aligned}$$

Let $S=G\times Y_{\min }$ and D be a generating set of G. Then the set

$$\begin{aligned} C=\{(g,1_Y)\mid g\in D\}\cup \{(1_G,y)\mid y\in Y\setminus \{1_Y\}\} \end{aligned}$$

is a generating set of S. Therefore, we have $\text {rank}(S)\le \text {rank}(G)+|Y|-1$. Since by the proof of Theorem 3.13 we have

$$\begin{aligned} \max _{S\in {\mathcal {M}}_n\cap {\mathcal {F}}}{|{{\mathcal {N}}}{{\mathcal {T}}}(S)\cap \mathrm{Gen} (S)|\over 2^{|S|-\text {rank}(S)}}\le & {} \max _{S\in {\mathcal {M}}_n\cap {\mathcal {F}}}{|{{\mathcal {N}}}{{\mathcal {T}}}(S)\cap \mathrm{Gen} (S)|\over 2^{|S|-\text {rank}(G)-|Y|+1}}\\\le & {} \max _{S\in {\mathcal {M}}_n\cap {\mathcal {F}}}{|{{\mathcal {N}}}{{\mathcal {T}}}(S)|\over 2^{|S|}}\times \left( 2^{\text {rank}(G)+|Y|-1}\right) \\\le & {} \max _{S\in {\mathcal {M}}_n\cap {\mathcal {F}}}{\left( {|G|^2\over 2^{|G|\over 4}}\right) ^{|Y|}}\times \left( 2^{(\log _2(|G|))^2+|Y|-1}\right) \\\le & {} \max _{S\in {\mathcal {M}}_n\cap {\mathcal {F}}}{\left( {2|G|^2\over 2^{|G|\over 4}}\right) ^{|Y|}}\times {|G|^2\over 2}, \end{aligned}$$

the result is clear.

((iii) $\Rightarrow$ (i)) For every monoid S, since $bg (S)\cap \mathrm{Gen} (S)\subseteq \mathrm{Norm} _U(S)\cap \mathrm{Gen} (S)$, the result is clear. $\square$

Finally, we state the origin of our counting method stated in Remak 3.3.

Remark 4.9

To state the relation between our method and topology and specially closure operators, first we need to recall some notions. Let A be a set. A mapping $\kappa : {\mathbb {P}}(A)\rightarrow {\mathbb {P}}(A)$ is called a closure operator on A if, for $X, Y\subseteq A$, it satisfies:

• C1   $X\subseteq \kappa (X)$

• C2    $\kappa (\kappa (X))=\kappa (X)$

• C3    $X\subseteq Y$ implies that $\kappa (X)\subseteq \kappa (Y)$.

A subset X of A is called a closed subset if $\kappa (X) = X$. The poset of closed subsets of A with set inclusion as the partial ordering is denoted by $L_\kappa$ (for more details, see [3]).

Now let $H\le \text {Aut}(S)$, where S is a monoid. Let $\kappa _H:{\mathbb {P}}(S)\rightarrow {\mathbb {P}}(S)$ be defined by $\kappa _H(X)=\cup _{x\in X} H\cdot x$, where $X\in {\mathbb {P}}(S)$. It is straightforward to see that $\kappa _H$ is a closure operator on S. Let H and $H'$ be subgroups of $\text {Aut}(S)$. Note that if H is a subgroup of $H'$, then $L_{\kappa _{H'}}\subseteq L_{\kappa _H}$.

Now note that for a subset C of a monoid S, if $|\text {Aut}(S,C)|\ne 1$, then $\text {Aut}(S,C)\cdot C=C$ and so $C\in L_{\text {Aut}(S,C)}$. On the other hand, note that there exists a minimal subgroup $P\in {\mathcal {P}}(S)$ such that $P\le \text {Aut}(S,C)$. So by the above discussion, $L_{\kappa _{\text {Aut}(S,C)}}\subseteq L_{\kappa _P}$. Hence, for every monoid S, we have

$$\begin{aligned} {{\mathcal {N}}}{{\mathcal {T}}}(S)=\cup _{P\in {\mathcal {P}}(S)}L_{\kappa _P}. \end{aligned}$$

On the other hand, note that for every $P\in {\mathcal {P}}(S)$, the set $L_{\kappa _P}$ is equal to N(P). This is the origin of our decomposition of ${{\mathcal {N}}}{{\mathcal {T}}}(S)$.