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Lattice-ordered pregroups are semi-distributive Algebra Univers. (IF 0.404) Pub Date : 2021-01-20 Nick Galatos, Peter Jipsen, Michael Kinyon, Adam Přenosil
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.
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Sandwiches for promise constraint satisfaction Algebra Univers. (IF 0.404) Pub Date : 2021-01-20 Guofeng Deng, Ezzeddine El Sai, Trevor Manders, Peter Mayr, Poramate Nakkirt, Athena Sparks
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
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Characterising certain compactifications of frames with special attention to Freudenthal Algebra Univers. (IF 0.404) Pub Date : 2021-01-19 Simo Mthethwa
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
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Neutrabelian algebras Algebra Univers. (IF 0.404) Pub Date : 2021-01-18 Keith A. Kearnes, Connor Meredith, Ágnes Szendrei
We introduce “neutrabelian algebras”, and prove that finite, hereditarily neutrabelian algebras with a cube term are dualizable.
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Solving a fixed number of equations over finite groups Algebra Univers. (IF 0.404) Pub Date : 2021-01-14 Philipp Nuspl
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
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Existence of cube terms in finite algebras Algebra Univers. (IF 0.404) Pub Date : 2021-01-12 Alexandr Kazda, Dmitriy Zhuk
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
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Mal’cev conditions corresponding to identities for compatible reflexive relations Algebra Univers. (IF 0.404) Pub Date : 2021-01-12 Stefano Fioravanti
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
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Sufficient conditions for the finiteness of the 3-generated lattice with modular and distributive type elements among generators Algebra Univers. (IF 0.404) Pub Date : 2021-01-07 Alexander G. Gein, Mikhail P. Shushpanov
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.
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Invariance of projective modules in $$\mathsf {Sup}$$ Sup under self-duality Algebra Univers. (IF 0.404) Pub Date : 2021-01-07 Javier Gutiérrez García, Ulrich Höhle, Tomasz Kubiak
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.
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Epimorphisms in varieties of subidempotent residuated structures Algebra Univers. (IF 0.404) Pub Date : 2021-01-04 T. Moraschini, J. G. Raftery, J. J. Wannenburg
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
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Results in L-algebras Algebra Univers. (IF 0.404) Pub Date : 2021-01-04 Lavinia Corina Ciungu
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
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Remarks about the Q-lattice of the variety of lattices Algebra Univers. (IF 0.404) Pub Date : 2021-01-04 M. E. Adams, W. Dziobiak
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
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The Tschantz and the alvin higher conditions are equivalent in congruence distributive varieties Algebra Univers. (IF 0.404) Pub Date : 2020-12-23 Paolo Lipparini
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.
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On Jónsson’s problem for groupoids of relations with operation of binary cylindrification Algebra Univers. (IF 0.404) Pub Date : 2020-11-16 Dmitry A. Bredikhin
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.
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Lattices from graph associahedra and subalgebras of the Malvenuto–Reutenauer algebra Algebra Univers. (IF 0.404) Pub Date : 2020-11-12 Emily Barnard, Thomas McConville
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
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Split epimorphisms as a productive tool in Universal Algebra Algebra Univers. (IF 0.404) Pub Date : 2020-11-10 Dominique Bourn
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.
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On the convergence rate of the fraction of simple algebras Algebra Univers. (IF 0.404) Pub Date : 2020-10-31 Florian Aichinger
We provide an improved lower bound for the convergence rate of the fraction of simple algebras using combinatorial arguments.
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Equidecomposable magmas Algebra Univers. (IF 0.404) Pub Date : 2020-10-30 Carles Cardó
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
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Properties not retained by pointed enrichments of finite lattices Algebra Univers. (IF 0.404) Pub Date : 2020-10-30 Ainur O. Basheyeva, Manat Mustafa, Anvar M. Nurakunov
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.
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Affine completeness of the algebra of full binary trees Algebra Univers. (IF 0.404) Pub Date : 2020-10-30 André Arnold, Patrick Cégielski, Serge Grigorieff, Irène Guessarian
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
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Congruence lattices of connected monounary algebras Algebra Univers. (IF 0.404) Pub Date : 2020-10-30 Danica Jakubíková-Studenovská, Lucia Janičková
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
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Lattice Burnside rings Algebra Univers. (IF 0.404) Pub Date : 2020-10-21 Fumihito Oda, Yugen Takegahara, Tomoyuki Yoshida
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
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Closed sets of finitary functions between finite fields of coprime order Algebra Univers. (IF 0.404) Pub Date : 2020-09-27 Stefano Fioravanti
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
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Some structural and residual properties of 2-semilattices Algebra Univers. (IF 0.404) Pub Date : 2020-09-13 Ian Payne
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
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Expanding Belnap: dualities for a new class of default bilattices Algebra Univers. (IF 0.404) Pub Date : 2020-09-09 Andrew P. K. Craig, Brian A. Davey, Miroslav Haviar
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Lattices do not distribute over powerset Algebra Univers. (IF 0.404) Pub Date : 2020-09-04 Julián Salamanca Téllez
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.
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On saturated varieties of posemigroups Algebra Univers. (IF 0.404) Pub Date : 2020-08-29 Shabir Ahmad Ahanger, Aftab Hussain Shah, Noor Mohammad Khan
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
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A note on the Knaster–Tarski Fixpoint Theorem Algebra Univers. (IF 0.404) Pub Date : 2020-08-09 Mengqiao Huang, Yuxi Fu
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.
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Representing subalgebras as retracts of finite subdirect powers Algebra Univers. (IF 0.404) Pub Date : 2020-08-06 Keith A. Kearnes, Alexander Rasstrigin
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 \(\mathbb A\).
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Lattice universality of locally finite 2-groups Algebra Univers. (IF 0.404) Pub Date : 2020-07-20 Vladimir B. Repnitskiǐ
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.
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Test sets for equality of terms in the additive structure of ordinals augmented with right multiplication by $$\omega $$ ω Algebra Univers. (IF 0.404) Pub Date : 2020-07-20 Christian Choffrut, Serge Grigorieff
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
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Filter and weak link topologies Algebra Univers. (IF 0.404) Pub Date : 2020-07-13 Mohammad A. Ranjbar, Mahmood Pourgholamhossein
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.
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On geometric posets and partial matroids Algebra Univers. (IF 0.404) Pub Date : 2020-07-13 Branimir Šešelja, Anna Slivková, Andreja Tepavčević
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
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Generalizing Galvin and Jónsson’s classification to $$N_5$$ N 5 Algebra Univers. (IF 0.404) Pub Date : 2020-07-13 Brian T. Chan
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
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Multiplayer rock–paper–scissors Algebra Univers. (IF 0.404) Pub Date : 2020-07-08 Charlotte Aten
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.
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The spectrum problem for Abelian $$\ell $$ ℓ -groups and MV-algebras Algebra Univers. (IF 0.404) Pub Date : 2020-07-06 Giacomo Lenzi, Antonio Di Nola
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
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The category of G-posets Algebra Univers. (IF 0.404) Pub Date : 2020-06-30 Frank Murphy-Hernandez
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.
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Essential adjunction of a strong unit to an archimedean lattice-ordered group Algebra Univers. (IF 0.404) Pub Date : 2020-06-22 Anthony W. Hager, Philip Scowcroft
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
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On semiconic idempotent commutative residuated lattices Algebra Univers. (IF 0.404) Pub Date : 2020-06-19 Wei Chen
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
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The structure of generalized BI-algebras and weakening relation algebras Algebra Univers. (IF 0.404) Pub Date : 2020-06-16 Nikolaos Galatos, Peter Jipsen
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
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Lifting, n -dimensional spectral resolutions, and n -dimensional observables Algebra Univers. (IF 0.404) Pub Date : 2020-06-07 Anatolij Dvurečenskij, Dominik Lachman
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
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Semilattice sums of algebras and Mal’tsev products of varieties Algebra Univers. (IF 0.404) Pub Date : 2020-05-20 C. Bergman, T. Penza, A. B. Romanowska
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
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Perfect locales and localic real functions Algebra Univers. (IF 0.404) Pub Date : 2020-05-17 Javier Gutiérrez García, Tomasz Kubiak, Jorge Picado
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
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A non-finitely based involution semigroup of order five Algebra Univers. (IF 0.404) Pub Date : 2020-05-14 Meng Gao, Wen Ting Zhang, Yan Feng Luo
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.
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Rings in which idempotents generate maximal or minimal ideals Algebra Univers. (IF 0.404) Pub Date : 2020-05-08 Themba Dube, Mojtaba Ghirati, Sajad Nazari, Ali Taherifar
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
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On epimorphisms of ordered algebras Algebra Univers. (IF 0.404) Pub Date : 2020-05-05 Nasir Sohail, Boža Tasić
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 \(\mathcal {F}\).
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Structure theorems for idempotent residuated lattices Algebra Univers. (IF 0.404) Pub Date : 2020-05-04 José Gil-Férez, Peter Jipsen, George Metcalfe
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
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The structure group of a non-degenerate effect algebra Algebra Univers. (IF 0.404) Pub Date : 2020-04-27 Carsten Dietzel, Wolfgang Rump
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
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Identical inclusions of semilattices Algebra Univers. (IF 0.404) Pub Date : 2020-04-21 G. Mashevitzky
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
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Finitary shadows of compact subgroups of $$S(\omega )$$S(ω) Algebra Univers. (IF 0.404) Pub Date : 2020-04-11 B. Majcher-Iwanow
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
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Lattices and quantales of ideals of semigroups and their preservation under Morita contexts Algebra Univers. (IF 0.404) Pub Date : 2020-04-10 Valdis Laan, László Márki, Ülo Reimaa
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.
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Free and cofree S -quantales Algebra Univers. (IF 0.404) Pub Date : 2020-04-02 Xin Zhang, Bin Zhao
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.
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Two-element structures modulo primitive positive constructability Algebra Univers. (IF 0.404) Pub Date : 2020-03-31 Manuel Bodirsky, Albert Vucaj
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
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On the structure of C -algebras through atomicity and if-then-else Algebra Univers. (IF 0.404) Pub Date : 2020-03-31 Gayatri Panicker, K. V. Krishna, Purandar Bhaduri
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
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Varieties of semiassociative relation algebras and tense algebras Algebra Univers. (IF 0.404) Pub Date : 2020-03-31 James M. Koussas, Tomasz Kowalski
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
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On difunctionality of class relations Algebra Univers. (IF 0.404) Pub Date : 2020-03-30 Michael Hoefnagel, Zurab Janelidze, Diana Rodelo
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
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Connected monads weakly preserve products Algebra Univers. (IF 0.404) Pub Date : 2020-03-30 H. Peter Gumm
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.
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Locales whose coz-complemented cozero sublocales have open closures Algebra Univers. (IF 0.404) Pub Date : 2020-03-29 Oghenetega Ighedo, Martin M. Mugochi
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
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Joins and Maltsev products of congruence permutable varieties Algebra Univers. (IF 0.404) Pub Date : 2020-02-25 Clifford Bergman
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.
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Notes on planar semimodular lattices. VIII. Congruence lattices of SPS lattices Algebra Univers. (IF 0.404) Pub Date : 2020-02-22 G. Grätzer
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
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