We prove that if \({\mathbb {A}}\) is an algebra that is supernilpotent with respect to the 2-term higher commutator, and \({\mathbb {B}}\) is a subalgebra of \({\mathbb {A}}\), then \({\mathbb {B}}\) is representable as a retract of a finite subdirect power of \(\mathbb A\).

We prove that every algebraic lattice is isomorphic to a complete sublattice in the subgroup lattice of a suitable locally finite 2-group. In particular, every lattice is embeddable in the subgroup lattice of a locally finite 2-group.

We consider the additive structure of ordinals augmented with right multiplication by \(\omega \). We prove that two terms in the algebra are semantically equal if and only if they take the same value on all elements of a set containing 0 and at least one ordinal of each finite degree. Among these (so-called) test sets the most natural one is that obtained by considering 0 along with all \(\omega \)-powers

In this paper we present the concept of weak link and weak link topology on lattice ordered groups and MV-algebras. We show that it is a locally solid group topology. In addition, we investigate the metrizability of a link topology.

The aim of this paper is to extend the notions of geometric lattices, semimodularity and matroids in the framework of finite posets and related systems of sets. We define a geometric poset as one which is atomistic and which satisfies particular conditions connecting elements to atoms. Next, by using a suitable partial closure operator and the corresponding partial closure system, we define a partial

The problem of determining (up to lattice isomorphism) the lattices that are sublattices of free lattices is in general an extremely difficult and an unsolved problem. A notable result towards solving this problem was established by Galvin and Jónsson when they classified (up to lattice isomorphism) all of the distributive sublattices of free lattices in 1959. In this paper, we weaken the requirement

We study a class of algebras we regard as generalized rock–paper–scissors games. We determine when such algebras can exist, show that these algebras generate the varieties generated by hypertournament algebras, count these algebras, study their automorphisms, and determine their congruence lattices. We produce a family of finite simple algebras.

This paper deals with the problem of characterizing those topological spaces which are homeomorphic to the prime spectra of MV-algebras or Abelian \(\ell \)-groups. As a first main result, we show that a topological space X is the prime spectrum of an MV-algebra if and only if X is spectral, and the lattice K(X) of compact open subsets of X is a closed epimorphic image of the lattice of “cylinder rational

For a group G and a poset X, we say that X is a G-poset if it is equipped with a G-action that is monotone. If we consider the mononote G-equivariant maps as morphisms, then we get the category of G-posets. We give a description of the projective objects, injective objects, quotients, and generators.

Within archimedean \(\ell \)-groups, “\(H \in W^{*}\)” means H contains a strong unit, and “\(G \in SW^{*}\) (respectively, \(ESW^{*}\))” means there are \(H \in W^{*}\) and an embedding (respectively, essential embedding) \(G \le H\). This paper continues earlier work in which we developed methods of attacking the question “\(G \in SW^{*}\)?” and gave many examples with answer “No” and “Yes”. Here

In this paper, we study semiconic idempotent commutative residuated lattices. An algebra of this kind is a semiconic generalized Sugihara monoid if it is generated by the lower bounds of the monoid identity. We establish a category equivalence between semiconic generalized Sugihara monoids and Brouwerian algebras with a strong nucleus. As an application, we show that central semiconic generalized Sugihara

Generalized bunched implication algebras (GBI-algebras) are defined as residuated lattices with a Heyting implication, and are positioned between Boolean algebras with operators and lattices with operators. We characterize congruences on GBI-algebras by filters that are closed under Gumm–Ursini terms, and for involutive GBI-algebras these terms simplify to a dual version of the congruence term for

We show that under some natural conditions, we are able to lift an n-dimensional spectral resolution from one monotone \(\sigma \)-complete unital po-group into another one, when the first one is a \(\sigma \)-homomorphic image of the second one. We note that an n-dimensional spectral resolution is a mapping from \(\mathbb R^n\) into a quantum structure which is monotone, left-continuous with non-negative

The Mal’tsev product of two varieties of similar algebras is always a quasivariety. We consider the question of when this quasivariety is a variety. The main result asserts that if \(\mathcal {V}\) is a strongly irregular variety with no nullary operations and at least one non-unary operation, and \(\mathcal {S}\) is the variety, of the same type as \(\mathcal {V}\), equivalent to the variety of semilattices

The purpose of this paper is to identify the role of perfectness in the Michael insertion theorem for perfectly normal locales. We attain it by characterizing perfect locales in terms of strict insertion of two comparable lower semicontinuous and upper semicontinuous localic real functions. That characterization, when combined with the insertion theorem for normal locales, provides an improved formulation

Since the 1970s, the smallest known non-finitely based involution semigroups are of order six. This paper exhibits the first example of a non-finitely based involution semigroup of order five.

We characterize rings in which every left ideal generated by an idempotent different from 0 and 1 is either a maximal left ideal or a minimal left ideal. In the commutative case, we give a characterization in terms of topological properties of the maximal spectrum with the Zariski topology. We also consider a strictly weaker variant of this property, defined almost similarly, and characterize those

We prove that epimorphisms are surjective in certain categories of ordered \(\mathcal {F}\)-algebras. It then turns out that epimorphisms are also surjective in the category of all (unordered) algebras of type \(\mathcal {F}\).

In this paper we study structural properties of residuated lattices that are idempotent as monoids. We provide descriptions of the totally ordered members of this class and obtain counting theorems for the number of finite algebras in various subclasses. We also establish the finite embeddability property for certain varieties generated by classes of residuated lattices that are conservative in the

A (non-commutative) structure group G(E) is associated to an arbitrary effect algebra E, and a concept of non-degeneracy is introduced. If E is non-degenerate, G(E) has a right invariant partial order, E embeds as an interval into G(E), and the negative cone of G(E) is a self-similar partial L-algebra. In the degenerate case, the possible anomalies are explained. Lattice effect algebras, 2-divisible

The class of identical inclusions was defined by Lyapin. We prove that any set of identical inclusions in the class of semilattices is equivalent to an elementary (the first order) formula. Elementary identical inclusions forms the class of universal formulas which is situated strictly between identities and universal positive formulas. We describe the infinite lattice of all classes of semilattices

Let LF be the lattice of all subgroups of the group \(SF(\omega )\) of all finitary permutations of the set of natural numbers. We consider subgroups of \(SF(\omega )\) of the form \(C\cap SF(\omega )\), where C is a compact subgroup of the group of all permutations. In particular, we study their distribution among elements of LF. We measure this using natural relations of orthogonality and almost

We study properties of the lattice of unitary ideals of a semigroup. In particular, we show that it is a quantale. We prove that if two semigroups are connected by an acceptable Morita context then there is an isomorphism between the quantales of unitary ideals of these semigroups. Moreover, factorisable ideals corresponding to each other under this isomorphism are strongly Morita equivalent.

In this paper, we give an alternative representation of a free S-quantale over a poset. Furthermore, we give the concrete forms of free and cofree S-quantales over a sup-lattice. Finally, we give an example to present that a cofree S-quantale over a poset does not always exist.

Primitive positive constructions have been introduced in recent work of Barto, Opršal, and Pinsker to study the computational complexity of constraint satisfaction problems. Let \({\mathfrak {P}}_{\mathrm {fin}}\) be the poset which arises from ordering all finite relational structures by pp-constructability. This poset is infinite, but we do not know whether it is uncountable. In this article, we

This paper investigates the notions of atoms and atomicity in C-algebras and obtains a characterisation of atoms in the C-algebra of transformations. In this connection, various characterisations for the existence of suprema of subsets of C-algebras are obtained. Further, this work presents some necessary conditions and some sufficient conditions for the atomicity of C-algebras and shows that the class

It is well known that the subvariety lattice of the variety of relation algebras has exactly three atoms. The (join-irreducible) covers of two of these atoms are known, but a complete classification of the (join-irreducible) covers of the remaining atom has not yet been found. These statements are also true of a related subvariety lattice, namely the subvariety lattice of the variety of semiassociative

For a given variety \({\mathcal {V}}\) of algebras, we define a class relation to be a binary relation \(R\subseteq S^2\) which is of the form \(R=S^2\cap K\) for some congruence class K on \(A^2\), where A is an algebra in \( {\mathcal {V}}\) such that \(S\subseteq A\). In this paper we study the following property of \({\mathcal {V}}\): every reflexive class relation is an equivalence relation. In

If F is a (not necessarily associative) monad on Set, then the natural transformation \(F(A\times B)\rightarrow F(A)\times F(B)\) is surjective if and only if \(F(\varvec{1})=\varvec{1}\). Specializing F to \(F_{\mathcal {V}}\), the free algebra functor for a variety \(\mathcal {V},\) this result generalizes and clarifies an observation by Dent, Kearnes and Szendrei in 2012.

A cozero sublocale C of a completely regular locale L will here be called coz-complemented if there is a cozero sublocale D of L such that \(C\cap D\) is the void sublocale and \(C\vee D\) is dense in L. Following terminology in spaces, we say L is a cloz-locale if every coz-complemented sublocale of L has an open closure. We characterize cloz-locales in terms of certain embeddings, and also in terms

Let \(\mathcal {A}\) and \(\mathcal {B}\) be idempotent varieties and suppose that the variety \(\mathcal {A} \vee \mathcal {B} \) is congruence permutable. Then the Maltsev product \(\mathcal {A} \circ \mathcal {B} \) is also congruence permutable.

In this note, I find a new property of the congruence lattice, \({{\,\mathrm{Con}\,}}L\), of an SPS lattice L (slim, planar, semimodular, where “slim” is the absence of \({\mathsf {M}}_3\) sublattices) with more than 2 elements: there are at least two dual atoms in\({{\,\mathrm{Con}\,}}L\). So the three-element chain cannot be represented as the congruence lattice of an SPS lattice, supplementing a result

We prove that an idempotent operation generates a loop from a strongly connected digraph containing directed closed walks of all lengths under very mild (local) algebraic assumptions. Using the result, we reprove the existence of weakest non-trivial idempotent equations, and that a finite strongly connected digraph of algebraic length 1 compatible with a Taylor operation has a loop.