Abstract: Let L be an n-element finite lattice. We prove that if L has more than \(2^{n-5}\) congruences, then L is planar. This result is sharp, since for each natural number \(n\ge 8\) , there exists a non-planar lattice with exactly \(2^{n-5}\) congruences. PubDate: 2019-03-02

Abstract: In an earlier article, the authors found sufficient conditions for complexity of the lattice of subquasivarieties of a quasivariety. In the present article, we prove that these conditions allow us to represent finite lattices as relative congruence lattices and relative variety lattices in a uniform way. Some applications are presented. PubDate: 2019-03-01

Abstract: We consider the problem of finding lower bounds on the number of unlabeled n-element lattices in some lattice family. We show that if the family is closed under vertical sum, exponential lower bounds can be obtained from vertical sums of small lattices whose numbers are known. We demonstrate this approach by establishing that the number of modular lattices is at least \(2.2726^n\) for n large enough. We also present an analogous method for finding lower bounds on the number of vertically indecomposable lattices in some family. For this purpose we define a new kind of sum, the vertical 2-sum, which combines lattices at two common elements. As an application we prove that the numbers of vertically indecomposable modular and semimodular lattices are at least \(2.1562^n\) and \(2.6797^n\) for n large enough. PubDate: 2019-02-22

Abstract: Up to isomorphism, there exist two non-isomorphic two-element monoids. We show that the identities of the free product of every pair of such monoids admit no finite basis. PubDate: 2019-02-22

Abstract: By a 1941 result of Ph. M. Whitman, the free lattice \({{\,\mathrm{FL}\,}}(3)\) on three generators includes a sublattice S that is isomorphic to the lattice \({{\,\mathrm{FL}\,}}(\omega )={{\,\mathrm{FL}\,}}(\aleph _0)\) generated freely by denumerably many elements. The first author has recently “symmetrized” this classical result by constructing a sublattice \(S\cong {{\,\mathrm{FL}\,}}(\omega )\) of \({{\,\mathrm{FL}\,}}(3)\) such that S is selfdually positioned in \({{\,\mathrm{FL}\,}}(3)\) in the sense that it is invariant under the natural dual automorphism of \({{\,\mathrm{FL}\,}}(3)\) that keeps each of the three free generators fixed. Now we move to the furthest in terms of symmetry by constructing a selfdually positioned sublattice \(S\cong {{\,\mathrm{FL}\,}}(\omega )\) of \({{\,\mathrm{FL}\,}}(3)\) such that every element of S is fixed by all automorphisms of \({{\,\mathrm{FL}\,}}(3)\) . That is, in our terminology, we embed \({{\,\mathrm{FL}\,}}(\omega )\) into \({{\,\mathrm{FL}\,}}(3)\) in a totally symmetric way. Our main result determines all pairs \((\kappa ,\lambda )\) of cardinals greater than 2 such that \({{\,\mathrm{FL}\,}}(\kappa )\) is embeddable into \({{\,\mathrm{FL}\,}}(\lambda )\) in a totally symmetric way. Also, we relax the stipulations on \(S\cong {{\,\mathrm{FL}\,}}(\kappa )\) by requiring only that S is closed with respect to the automorphisms of \({{\,\mathrm{FL}\,}}(\lambda )\) , or S is selfdually positioned and closed with respect to the automorphisms; we determine the corresponding pairs \((\kappa ,\lambda )\) even in these two cases. We reaffirm some of our calculations with a computer program developed by the first author. This program is for the word problem of free lattices, it runs under Windows, and it is freely available. PubDate: 2019-02-07

Abstract: We investigate the alternate order on a congruence-uniform lattice \({\mathcal {L}}\) as introduced by N. Reading, which we dub the core label order of \({\mathcal {L}}\) . When \({\mathcal {L}}\) can be realized as a poset of regions of a simplicial hyperplane arrangement, the core label order is always a lattice. For general \({\mathcal {L}}\) , however, this fails. We provide an equivalent characterization for the core label order to be a lattice. As a consequence we show that the property of the core label order being a lattice is inherited to lattice quotients. We use the core label order to characterize the congruence-uniform lattices that are Boolean lattices, and we investigate the connection between congruence-uniform lattices whose core label orders are lattices and congruence-uniform lattices of biclosed sets. PubDate: 2019-02-05

Abstract: It is known that any finite idempotent algebra that satisfies a nontrivial Maltsev condition must satisfy the linear one-equality Maltsev condition (a variant of the term discovered by Siggers and refined by Kearnes, Marković, and McKenzie): $$\begin{aligned} t(r,a,r,e)\approx t(a,r,e,a). \end{aligned}$$ We show that if we drop the finiteness assumption, the k-ary weak near unanimity equations imply only trivial linear one-equality Maltsev conditions for every \(k\ge 3\) . From this it follows that there is no nontrivial linear one-equality condition that would hold in all idempotent algebras having Taylor terms. Miroslav Olšák has recently shown that there is a weakest nontrivial strong Maltsev condition for idempotent algebras. Olšák has found several such (mutually equivalent) conditions consisting of two or more equations. Our result shows that Olšák’s equation systems cannot be compressed into just one equation. PubDate: 2019-02-04

Abstract: In 1998 the author showed that the canonical extension of a bounded modular lattice need not be modular. The proof was indirect, using a deep result of Kaplansky. In this note we give an explicit example. PubDate: 2019-02-04

Abstract: The dualisability of partial algebras is a largely unexplored area within natural duality theory. This paper considers the dualisability of finite structures that have a single partial unary operation in the type. We show that every such finite partial unar is dualisable. We obtain this result by showing that the relational structure obtained by replacing the fundamental operation by its graph is dualisable. We also give a finite generator for the class of all disjoint unions of directed trees up to some fixed height, considered as partial unars. PubDate: 2019-01-30

Abstract: We show that a von Neumann regular ring with involution is directly finite provided that it admits a representation as a \(*\) -ring of endomorphisms of a vector space endowed with a non-degenerate orthosymmetric sesquilinear form. PubDate: 2019-01-29

Abstract: We consider the problem of approximating distributions of Bernoulli random variables by applying Boolean functions to independent random variables with distributions from a given set. For a set B of Boolean functions, the set of approximable distributions forms an algebra, named the approximation algebra of Bernoulli distributions induced by B. We provide a complete description of approximation algebras induced by most clones of Boolean functions. For remaining clones, we prove a criterion for approximation algebras and a property of algebras that are finitely generated. PubDate: 2019-01-29

Abstract: We prove that Mal’tsev and Goursat categories may be characterized through variations of the Shifting Lemma, that is classically expressed in terms of three congruences R, S and T, and characterizes congruence modular varieties. We first show that a regular category \({\mathbb {C}}\) is a Mal’tsev category if and only if the Shifting Lemma holds for reflexive relations on the same object in \({\mathbb {C}}\) . Moreover, we prove that a regular category \({\mathbb {C}}\) is a Goursat category if and only if the Shifting Lemma holds for a reflexive relation S and reflexive and positive relations R and T in \({\mathbb {C}}\) . In particular this provides a new characterization of 2-permutable and 3-permutable varieties and quasi-varieties of universal algebras. PubDate: 2019-01-09

Abstract: We establish a characterization of supernilpotent Mal’cev algebras which generalizes the affine structure of abelian Mal’cev algebras and the recent characterization of 2-supernilpotent Mal’cev algebras. We then show that for varieties in which the two-generated free algebra is finite: (1) neutrality of the higher commutators is equivalent to congruence meet-semidistributivity, and (2) the class of varieties which interpret a Mal’cev term in every supernilpotent algebra is equivalent to the existence of a weak difference term. We then establish properties of the higher commutator in the aforementioned second class of varieties. PubDate: 2018-12-19

Abstract: Given a dense additive subgroup G of \(\mathbb {R}\) containing \(\mathbb {Z}\) , we consider its intersection \(\mathbb {G}\) with the interval [0, 1[ with the induced order and the group structure given by addition modulo 1. We axiomatize the theory of \(\mathbb {G}\) and show it is model-complete, using a Feferman-Vaught type argument. We show that any sufficiently saturated model decomposes into a product of a standard part and two ordered semigroups of infinitely small and infinitely large elements. PubDate: 2018-11-27

Abstract: Although there have been repeated attempts to define the concept of an Archimedean algebra for individual classes of residuated lattices, there is no all-purpose definition that suits the general case. We suggest as a possible candidate the notion of a normal-valued and e-cyclic residuated lattice that has the zero radical compact property—namely, a normal-valued and e-cyclic residuated lattice in which every principal convex subuniverse has a trivial radical (understood as the intersection of all its maximal convex subuniverses). We characterize the Archimedean members in the variety of e-cyclic residuated lattices, as well as in various special cases of interest. A theorem to the effect that each Archimedean and prelinear GBL-algebra is commutative, subsuming as corollaries several analogous results from the recent literature, is grist to the mill of our proposal’s adequacy. Finally, we revisit the concept of a hyper-Archimedean residuated lattice, another notion with which researchers have engaged from disparate angles, and investigate some of its properties. PubDate: 2018-11-24

Abstract: The productivity of the \(\kappa \) -chain condition, where \(\kappa \) is a regular, uncountable cardinal, has been the focus of a great deal of set-theoretic research. In the 1970s, consistent examples of \(\kappa \) -cc posets whose squares are not \(\kappa \) -cc were constructed by Laver, Galvin, Roitman and Fleissner. Later, \(\textsf {ZFC}\) examples were constructed by Todorcevic, Shelah, and others. The most difficult case, that in which \(\kappa = \aleph _2\) , was resolved by Shelah in 1997. In this work, we obtain analogous results regarding the infinite productivity of strong chain conditions, such as the Knaster property. Among other results, for any successor cardinal \(\kappa \) , we produce a \(\textsf {ZFC}\) example of a poset with precaliber \(\kappa \) whose \(\omega ^{\mathrm {th}}\) power is not \(\kappa \) -cc. To do so, we carry out a systematic study of colorings satisfying a strong unboundedness condition. We prove a number of results indicating circumstances under which such colorings exist, in particular focusing on cases in which these colorings are moreover closed. PubDate: 2018-11-23

Abstract: This paper is a continuation of the earlier paper by the same authors in which a primary result was that every arithmetical affine complete variety of finite type is a principal arithmetical variety with respect to an appropriately chosen Pixley term. The paper begins by presenting an extension of this result to all finitely generated congruences and, as an example, constructs a closed form solution formula for any finitely presented system of pairwise compatible congruences (the Chinese remainder theorem). It is also shown that in all such varieties the meet of principal congruences is also principal, and finally, if a minimal generating algebra of the variety is regular, it is shown that the variety is also regular and the join of principal congruences is again principal. PubDate: 2018-11-21