It is proven in this note that if non-empty posets A, B, C, and D satisfy \(A^C\cong B^D\) where C, D, and \(A^C\) are finite and connected, then there exist posets E, X, Y, and Z such that \(A\cong E^X\), \(B\cong E^Y\), \(C\cong Y\times Z\), and \(D\cong X\times Z\). This solves a problem posed by Jónsson and McKenzie in 1982.

We investigate the computational complexity of the problem of deciding if an algebra homomorphism can be factored through an intermediate algebra. Specifically, we fix an algebraic language, \(\mathcal L\), and take as input an algebra homomorphism \(f:X\rightarrow Z\) between two finite \(\mathcal L\)-algebras X and Z, along with an intermediate finite \(\mathcal L\)-algebra Y. The decision problem

An algebra-valued model of set theory is called loyal to its algebra if the model and its algebra have the same propositional logic; it is called faithful if all elements of the algebra are truth values of a sentence of the language of set theory in the model. We observe that non-trivial automorphisms of the algebra result in models that are not faithful and apply this to construct three classes of

We provide a generalization of Mundici’s equivalence between unital Abelian lattice-ordered groups and MV-algebras: the category of unital commutative lattice-ordered monoids is equivalent to the category of MV-monoidal algebras. Roughly speaking, unital commutative lattice-ordered monoids are unital Abelian lattice-ordered groups without the unary operation \(x \mapsto -x\). The primitive operations

Let \(\mathbb {S}\) be the commutative and idempotent semiring with additive identity \(\mathbf {0}\) and multiplicative identity \(\mathbf {1}\). The tropical semiring \(\mathbb {T}\) and the Boolean semiring \(\mathbb {B}\) are common important examples of such semirings. Let \(UT_{n}(\mathbb {S})\) be the semigroup of all \(n\times n\) upper triangular matrices over \(\mathbb {S}\), both \(UT^{\pm

We are interested in abstract conditions that characterize homomorphic images of affine quandles. Our main result is a two-fold characterization of this class: one by a property of the displacement group, the other one by a property of the corresponding affine mesh. As a consequence, we obtain efficient algorithms for recognizing homomorphic images of affine quandles, including an efficient explicit

The 6-element Brandt monoid \(B_2^1\) admits a unique addition under which it becomes an additively idempotent semiring. We show that this addition is a term operation of \(B_2^1\) as an inverse semigroup. As a consequence, we exhibit an easy proof that the semiring identities of \(B_2^1\) are not finitely based.

Let \(\sum (X)\) be the collection of subrings of C(X) containing \(C^{*}(X)\), where X is a Tychonoff space. For any \(A(X)\in \sum (X)\) there is associated a subset \(\upsilon _{A}(X)\) of \(\beta X\) which is an A-analogue of the Hewitt real compactification \(\upsilon X\) of X. For any \(A(X)\in \sum (X)\), let [A(X)] be the class of all \(B(X)\in \sum (X)\) such that \(\upsilon _{A}(X)=\upsilon

The paper deals with subnormal and composition subgroups in the framework of weak congruence lattices of groups. Weak congruences of the composition subgroups of a group form a sublattice of the lattice of all weak congruences. We characterize normality and subnormality in purely lattice-theoretic terms. For a finite group G we prove: all subgroups of G are subnormal (i.e., G is nilpotent) if and only

In this paper, we find axiom systems and bases of identities for classes of algebras of binary relations with operations defined by logical formulas containing only conjunctions.

A relational structure is called reversible iff every bijective endomorphism of that structure is an automorphism. We characterize that property in the class of disconnected binary structures, and in one of its subclasses—posets that are disjoint union of chains. Also, using the concept of a monotone function from a class of structures preordered by monomorphism into a well founded relation, we detect

We introduce notions of Galois biconnections, intended to be the bilattice analogue of classical Galois connections between lattices. A bidirectional Galois biconnection is a (compatible) pair of Galois connections, the first relating truth orderings and the second relating knowledge orderings, while a unidirectional Galois biconnection is a Galois connection equipped with extra properties that seek

Let \((\dagger )\) denote the following property of a variety \(\mathcal V\): Every subquasivariety of \(\mathcal V\) is a variety. In this paper, we prove that every idempotent dual discriminator variety has property \((\dagger )\). Property \((\dagger )\) need not hold for nonidempotent dual discriminator varieties, but \((\dagger )\) does hold for minimal nonidempotent dual discriminator varieties

We unify several extensions of the classic Stone duality due to Grätzer, Hofmann–Lawson and Jung–Sünderhauf. Specifically, we show that \(\cup \)-bases of locally compact sober spaces are dual to \(\prec \)-distributive \(\vee \)-predomains, where \(\prec \) is a relation representing compact containment.

In this paper we discuss the relationship between direct products of monounary algebras and their components, with respect to the properties of residual finiteness, strong/weak subalgebra separability, and complete separability. For each of these properties \({\mathcal {P}}\), we give a criterion \(\mathcal {C_P}\) such that a monounary algebra \(A\) has property \({\mathcal {P}}\) if and only if it

We investigate dual spaces of congruence lattices of algebras in a congruence-distributive variety \({\mathcal V}\). Our aim is to connect topological properties of these spaces with diagrams of finite \((\vee ,0)\)-semilattices liftable in \({\mathcal V}\). We achieve this aim for diagrams indexed by finite trees.

We investigate the Mal’tsev product \(\mathcal {V}\circ \mathcal {W}\) of two varieties \(\mathcal {V}\) and \(\mathcal {W}\) of the same similarity type. While such a product is usually a quasivariety, it is not necessarily a variety. We give an equational base for the variety generated by \(\mathcal {V}\circ \mathcal {W}\) in terms of identities satisfied in \(\mathcal {V}\) and \(\mathcal {W}\)

In this paper, we show that the category of \(\kappa \)-additive complete atomic modal algebras is dually equivalent to the category of \(\kappa \)-downward directed multi-relational Kripke frames, for any cardinal number \(\kappa \). Multi-relational Kripke frames are not Kripke frames for multi-modal logics, but frames for monomodal logics in which the modal operator \(\Diamond \) does not distribute

We study clones on a four-element set related to the clone \({\mathsf {DMA}}\) of all term functions of the subdirectly irreducible four-element De Morgan algebra \({\mathbf {DM}}_{{{\mathbf {4}}}}\). We find generating sets for the clones of all functions preserving the subalgebras of \({\mathbf {DM}}_{{{\mathbf {4}}}}\), the automorphisms of \({\mathbf {DM}}_{{{\mathbf {4}}}}\), the truth order and

In this paper, we characterize pseudocomplemented lattices with (ACC) and (DCC) by using their sublattices. Specifically speaking, we give a necessary and sufficient condition that a lattice with (ACC) and (DCC) is pseudocomplemented by using its nine sublattices. Moreover, we describe the interval pseudocomplementedness of semimodular lattices with (ACC) and (DCC) by their sublattices equivalently

The motivation for using demonic calculus for binary relations stems from the behaviour of demonic turing machines, when modelled relationally. Relational composition (; ) models sequential runs of two programs and demonic refinement (\(\sqsubseteq \)) arises from the partial order given by modeling demonic choice (\(\sqcup \)) of programs (see below for the formal relational definitions). We prove

We give axioms for a class of ordered structures, called truncated ordered abelian groups (TOAG’s) carrying an addition. TOAG’s come naturally from ordered abelian groups with a 0 and a \(+\), but the addition of a TOAG is not necessarily even a cancellative semigroup. The main examples are initial segments \([0, \tau ]\) of an ordered abelian group, with a truncation of the addition. We prove that

This note aims to clarify the relations between three ways of constructing complete lattices that appear in three different areas: (1) using ordered structures, as in set-theoretic forcing, or doubly ordered structures, as in a recent semantics for intuitionistic logic; (2) using compatibility relations, as in semantics for quantum logic based on ortholattices; (3) using Birkhoff’s polarities, as in

We prove that for every \(n>2\) there exists an o-group G of finite Archimedean rank n such that G cannot be embedded in a divisible o-group of Archimedean rank n, and also prove that G can be embedded in an o-group of finite Archimedean rank greater than n.

Demonic composition \(*\) is an associative operation on binary relations, and demonic refinement \({\sqsubseteq }\) is a partial order on binary relations. Other operations on binary relations considered here include the unary domain operation D and the left restrictive multiplication operation \(\circ \) given by \(s\circ t=D(s)*t\). We show that the class of relation algebras of signature \(\{\

In this paper, we study principal element lattices, Prüfer lattices and Q-lattices. Also, a necessary and sufficient condition is given for a principally generated C-lattice to be a finite direct product of proper Dedekind domains.

We explore a non-abelian torsion theory in the category of preordered groups: the objects of its torsion-free subcategory are the partially ordered groups, whereas the objects of the torsion subcategory are groups (with the total order). The reflector from the category of preordered groups to this torsion-free subcategory has stable units, and we prove that it induces a monotone-light factorization

G. Czédli proved that the blocks of any compatible tolerance T of a lattice L can be ordered in such a way that they form a lattice L/T called the factor lattice of L modulo T. Here we show that the Dedekind–MacNeille completion of the lattice L/T is isomorphic to the concept lattice of the context (L, L, R), where R stands for the reflexive weak ordered relation \( \mathord {\le } \circ T\). Weak

Classes of multisorted minions closed under extensions, reflections, and direct powers are considered from a relational point of view. As a generalization of a result of Barto, Opršal, and Pinsker, the closure of a multisorted minion is characterized in terms of constructions on multisorted relation pairs which are invariant for minions.

For a variety \({\mathcal {V}}\), it has been recently shown that binary products commute with arbitrary coequalizers locally, i.e., in every fibre of the fibration of points \(\pi : \mathrm {Pt}({\mathbb {C}}) \rightarrow {\mathbb {C}}\), if and only if Gumm’s shifting lemma holds on pullbacks in \({\mathcal {V}}\). In this paper, we establish a similar result connecting the so-called triangular lemma

New examples of existentially closed Abelian lattice-ordered groups, possessing most of the known properties of infinitely generic Abelian lattice-ordered groups, are constructed using Fraïssé limits.

In this note, we derive a criterion for determining when two groups generate the same group variety. We use this, along with a description of the 2-variable word maps on the dihedral, dicyclic, and semi-dihedral groups, to show that the maximal class 2-groups of the same order generate the same variety.

We prove that the lattice reduct of every lattice-ordered pregroup is semidistributive. This is a consequence of a certain weak form of the distributive law which holds in lattice-ordered pregroups.

Promise Constraint Satisfaction Problems (\(\mathrm{PCSP}\)) were proposed recently by Brakensiek and Guruswami as a framework to study approximations for Constraint Satisfaction Problems (\(\mathrm{CSP}\)). Informally a \(\mathrm{PCSP}\) asks to distinguish between whether a given instance of a \(\mathrm{CSP}\) has a solution or not even a specified relaxation can be satisfied. All currently known

We introduce the definition of h-perfect elements relative to a compactification \(h:M\longrightarrow L\) and show that if a collection of all such elements is a basis, then the remainder of a frame in this compactification is zero-dimensional. This concept yields what we call a full \(\pi \)-compact basis for rim-compact frames. Compactifications arising from full \(\pi \)-compact bases are investigated

We introduce “neutrabelian algebras”, and prove that finite, hereditarily neutrabelian algebras with a cube term are dualizable.

We investigate the complexity of solving systems of polynomial equations over finite groups. In 1999 Goldmann and Russell showed \(\mathrm {NP}\)-completeness of this problem for non-Abelian groups. We show that the problem can become tractable for some non-Abelian groups if we fix the number of equations. Recently, Földvári and Horváth showed that a single equation over groups which are semidirect

We study the problem of whether a given finite algebra with finitely many basic operations contains a cube term; we give both structural and algorithmic results. We show that if such an algebra has a cube term then it has a cube term of dimension at most N, where the number N depends on the arities of basic operations of the algebra and the size of the basic set. For finite idempotent algebras we give

We investigate Mal’cev conditions described by those equations whose variables runs over the set of all compatible reflexive relations. Let \(p \le q\) be an equation for the language \(\{\wedge , \circ ,+\}\). We give a characterization of the class of all varieties which satisfy \(p \le q\) over the set of all compatible reflexive relations. The aim is to find an analogon of the Pixley–Wille algorithm

We investigate the finiteness and the infiniteness of the free 3-generated lattice with modular and distributive type elements among generators. For all triples of generators with modular and distributive type elements, it is discovered whether the lattice generated by such triple is finite or infinite.

Starting from enriched order-theoretic properties of modules over a unital quantale in the category \(\mathsf {Sup}\), this paper presents the following theorem. If the underlying quantale is unital and involutive with a designated element, then the duality of right (left) modules preserves projectivity if and only if the underlying quantale has a dualizing element.

A commutative residuated lattice \({\varvec{A}}\) is said to be subidempotent if the lower bounds of its neutral element e are idempotent (in which case they naturally constitute a Brouwerian algebra \({\varvec{A}}^-\)). It is proved here that epimorphisms are surjective in a variety \({\mathsf {K}}\) of such algebras \({\varvec{A}}\) (with or without involution), provided that each finitely subdirectly

We study some classes of L-algebras and we give characterizations of commutative KL-algebras and CL-algebras. It is proved that any commutative KL-algebra is a BCK-algebra and any self-similar CL-algebra is commutative. The commutative ideals are defined and studied, and it is proved that a CL-algebra is commutative if and only if all its ideals are commutative. We also present the L-algebras arising

Let L denote the Q-lattice of the variety \({\mathcal {V}}\) of lattices, i.e. the lattice of quasivarieties that are contained in \({\mathcal {V}}\). Let F denote the free lattice in \({\mathcal {V}}\) with \(\omega \) free generators and let Q(F) denote the quasivariety of lattices generated by F. Let Fin denote the collection of finite lattices which belong to Q(F) and let Q(Fin) denote the quasivariety

We show that, under the assumption of congruence distributivity, a condition by S. Tschantz characterizing congruence modularity is equivalent to a variant of the classical Jónsson condition. Here equivalence is intended in a strong sense, to the effect that the corresponding sequences of terms have exactly the same length.

In this paper, it is proved that the quasi-variety generated by the class of partially ordered groupoids of relations with the operation of binary cylindrification forms a variety.

The Malvenuto–Reutenauer algebra is a well-studied combinatorial Hopf algebra with a basis indexed by permutations. This algebra contains a wide variety of interesting sub Hopf algebras, in particular the Hopf algebra of plane binary trees introduced by Loday and Ronco. We compare two general constructions of subalgebras of the Malvenuto–Reutenauer algebra, both of which include the Loday–Ronco algebra

By various examples, we show how, in Universal Algebra, the choice of a class of split epimorphisms \(\Sigma \), defined by specific equations or local operations in their fibres, can be used as a productive and flexible tool determining \(\Sigma \)-partial properties. We focus, here, our attention on \(\Sigma \)-partial congruence modular and \(\Sigma \)-partial congruence distributive formulae.

We provide an improved lower bound for the convergence rate of the fraction of simple algebras using combinatorial arguments.

A magma is called equidecomposable when the operation is injective, or, in other words, if \(x+y=x'+y'\) implies that \(x=x'\) and \(y=y'\). A magma is free iff it is equidecomposable and graded, hence the notion of equidecomposability is very related to the notion of freeness although it is not sufficient. We study main properties of such magmas. In particular, an alternative characterization of freeness

The present work considers algebras and their enrichments. It is shown by two examples of finite lattices that the properties “to have (no) a finite basis of quasi-identities” and “to generate a standard topological quasivariety” are not preserved with respect to pointed enrichments of finite algebras.

A function on an algebra is congruence preserving if, for any congruence, it maps pairs of congruent elements onto pairs of congruent elements. We show that on the algebra of full binary trees whose leaves are labeled by letters of an alphabet containing at least three letters, a function is congruence preserving if and only if it is polynomial. This exhibits an example of a non commutative and non

The system of all congruences of an algebra (A, F) forms a lattice, denoted \({{\,\mathrm{Con}\,}}(A, F)\). Further, the system of all congruence lattices of all algebras with the base set A forms a lattice \(\mathcal {E}_A\). We deal with meet-irreducibility in \(\mathcal {E}_A\) for a given finite set A. All meet-irreducible elements of \(\mathcal {E}_A\) are congruence lattices of monounary algebras

Given a finite group G and a finite G-lattice \({{\mathscr {L}}}\), we introduce the concept of lattice Burnside ring associated to a family of nonempty sublattices \({{\mathscr {L}}}_H\) of \({{\mathscr {L}}}\) for \(H\le G\). The slice Burnside ring introduced by Bouc is isomorphic to a lattice Burnside ring. Any lattice Burnside ring is an extension of the ordinary Burnside ring and is isomorphic

We investigate the finitary functions from a finite field \(\mathbb {F}_q\) to the finite field \(\mathbb {F}_p\), where p and q are powers of different primes. An \((\mathbb {F}_p,\mathbb {F}_q)\)-linearly closed clonoid is a subset of these functions which is closed under composition from the right and from the left with linear mappings. We give a characterization of these subsets of functions through

To each 2-semilattice, one can associate a digraph and a partial order. We analyze these two structures working toward two main goals: One goal is to give a structural dichotomy on minimal congruences of 2-semilattices. From this, we are able to deduce information about the tame-congruence-theoretic types that occur in 2-semilattices. In particular, we show that the type of a finite simple 2-semilattice

We show that there is no distributive law of the free lattice monad over the powerset monad. The proof presented here also works for other classes of lattices such as (bounded) distributive/modular lattices and also for some variants of the powerset monad such as the (nonempty) finite powerset monad.

We show that a permutative variety of posemigroups satisfying a permutation identity \(x_1x_2\cdots x_n=x_{i_1}x_{i_2}\cdots x_{i_n}\) with \(i_1\ne 1~ \text{ and }~i_{n-1}\ne n-1 ~[i_n\ne n~\text{ and }~i_{2}\ne 2]\) is saturated if and only if it admits an identity I such that I is not a permutation identity and at least one side of I has no repeated variables. Then we show that the variety of po-rectangular

This note shows that several statements about fixpoints in order theory are equivalent to Knaster–Tarski Fixpoint Theorem for complete lattices. All proofs have been done in Zermelo–Fraenkel set theory without the Axiom of Choice.