Authors:Graham Manuell Pages: 125 - 130 Abstract: Abstract J. Madden has shown that in contrast to the situation with frames, the smallest dense quotient of a \({\kappa}\) -frame need not be Boolean. We characterise these so-called \({d}\) -reduced \({\kappa}\) -frames as those which may be embedded as a generating sub- \({\kappa}\) -frame of a Boolean frame. We introduce the notion of the closure of a \({\kappa}\) -frame congruence and call a congruence clear if it is the largest congruence with a given closure. These ideas are used to prove \({\kappa}\) -frame analogues of known results concerning Boolean frame quotients. In particular, we show that d-reduced \({\kappa}\) -frames are precisely the quotients of \({\kappa}\) -frames by clear congruences and that every \({\kappa}\) -frame congruence is the meet of clear congruences. PubDate: 2017-10-01 DOI: 10.1007/s00012-017-0439-y Issue No:Vol. 78, No. 2 (2017)

Authors:Edmond W. H. Lee Pages: 131 - 145 Abstract: Abstract A finite algebra is completely join prime if whenever it belongs to the complete join of some collection of pseudovarieties, then it belongs to one of the pseudovarieties. An infinite class of completely join prime J-trivial semigroups with unique involution is introduced to demonstrate the incompatibility between the lattice of pseudovarieties of involution semigroups and the lattice of pseudovarieties of semigroups. Examples are also exhibited to show that a finite involution semigroup and its semigroup reduct need not be simultaneously completely join prime. PubDate: 2017-10-01 DOI: 10.1007/s00012-017-0442-3 Issue No:Vol. 78, No. 2 (2017)

Authors:Greg Oman Pages: 147 - 157 Abstract: Abstract Let S and T be sets with S infinite, and \({*:S \times S \rightarrow T}\) a function. Further, suppose that λ is a cardinal such that \({\aleph_0 \leq \lambda \leq S }\) . Say that (S, T, *) is elementarily λ-homogeneous provided (X, T,*) is elementarily equivalent to (Y, T, *) for all subsets X and Y of S of cardinality λ. In this note, we classify the elementarily λ-homogeneous structures (S, T, *). As corollaries, we characterize certain mathematical structures \({\mathfrak{S}}\) which are also “elementarily \({\lambda}\) -homogeneous” in the sense that all substructures of \({\mathfrak{S}}\) of cardinality λ are elementarily equivalent. Among our corollaries is a generalization of a theorem due to Manfred Droste. PubDate: 2017-10-01 DOI: 10.1007/s00012-017-0448-x Issue No:Vol. 78, No. 2 (2017)

Authors:G. Grätzer Abstract: Abstract A new result of G. Czédli states that for an ordered set P with at least two elements and a group G, there exists a bounded lattice L such that the ordered set of principal congruences of L is isomorphic to P and the automorphism group of L is isomorphic to G. I provide an alternative proof utilizing a result of mine with J. Sichler from the late 1990s. PubDate: 2017-10-17 DOI: 10.1007/s00012-017-0462-z

Authors:Friedrich Martin Schneider Abstract: Abstract Birkhoff’s HSP theorem characterizes the classes of models of algebraic theories as those being closed with respect to homomorphic images, subalgebras, and products. In particular, it implies that an algebra B satisfies all equations that hold in an algebra A of the same signature if and only if B is a homomorphic image of a subalgebra of a (possibly infinite) direct power of A. The former statement is equivalent to the existence of a natural map sending term functions of the algebra A to those of B—the natural clone homomorphism. The study of continuity properties of natural clone homomorphisms has been initiated recently by Bodirsky and Pinsker for locally oligomorphic algebras. Revisiting the argument of Bodirsky and Pinsker, we show that for any algebra B in the variety generated by an algebra A, the induced natural clone homomorphism is uniformly continuous if and only if every finitely generated subalgebra of B is a homomorphic image of a subalgebra of a finite power of A. Based on this observation, we study the question as to when Cauchy continuity of natural clone homomorphisms implies uniform continuity. We introduce the class of almost locally finite algebras, which encompasses all locally oligomorphic as well as all locally finite algebras, and show that, in case A is almost locally finite, then the considered natural homomorphism is uniformly continuous if (and only if) it is Cauchy-continuous. In particular, this provides a locally finite counterpart of the result by Bodirsky and Pinsker. Along the way, we also discuss some peculiarities of oligomorphic permutation groups on uncountable sets. PubDate: 2017-10-17 DOI: 10.1007/s00012-017-0460-1

Authors:Riquelmi Cardona; Nikolaos Galatos Abstract: Abstract We prove the finite embeddability property for a wide range of varieties of fully distributive residuated lattices and FL-algebras. Part of the axiomatization is assumed to be a knotted inequality and some appropriate generalization of commutativity. The construction is based on distributive residuated frames and extends to subvarieties axiomatized by any division-free equation. PubDate: 2017-10-17 DOI: 10.1007/s00012-017-0466-8

Authors:Mikhail P. Shushpanov Abstract: Abstract We consider a lattice generated by three elements, one of which is completely modular. The free lattice with this property is proved to be finite. It is not modular and contains exactly 39 elements. We have also found a finite set of defining relations for the generating elements of this lattice. PubDate: 2017-10-17 DOI: 10.1007/s00012-017-0463-y

Authors:G. Grätzer Abstract: Abstract A recent result of G. Czédli relates the ordered set of principal congruences of a bounded lattice L with the ordered set of principal congruences of a bounded sublattice K of L. In this note, I sketch a new proof. PubDate: 2017-10-17 DOI: 10.1007/s00012-017-0461-0

Authors:Gary L. Peterson Abstract: Abstract We correct a theorem in the original paper concerning when a composition series of a compatible nearring module is a 1-affine complete chain. PubDate: 2017-10-11 DOI: 10.1007/s00012-017-0458-8

Authors:Nikolaos Galatos; Peter Jipsen Abstract: Abstract We show that all extensions of the (non-associative) Gentzen system for distributive full Lambek calculus by simple structural rules have the cut elimination property. Also, extensions by such rules that do not increase complexity have the finite model property, hence many subvarieties of the variety of distributive residuated lattices have decidable equational theories. For some other extensions, we prove the finite embeddability property, which implies the decidability of the universal theory, and we show that our results also apply to generalized bunched implication algebras. Our analysis is conducted in the general setting of residuated frames. PubDate: 2017-10-09 DOI: 10.1007/s00012-017-0456-x

Authors:Miguel A. Campercholi; James G. Raftery Abstract: Abstract Quasivarietal analogues of uniform congruence schemes are discussed, and their relationship with the equational definability of principal relative congruences (EDPRC) is established, along with their significance for relative congruences on subalgebras of products. Generalizing the situation in varieties, we prove that a quasivariety is relatively ideal iff it has EDPRC; it is relatively filtral iff it is relatively semisimple with EDPRC. As an application, it is shown that a finitary sentential logic, algebraized by a quasivariety K, has a classical inconsistency lemma if and only if K is relatively filtral and the subalgebras of its nontrivial members are nontrivial. A concrete instance of this result is exhibited, in which K is not a variety. Finally, for quasivarieties \({\sf{M} \subseteq \sf{K}}\) , we supply some conditions under which M is the restriction to K of a variety, assuming that K has EDPRC. PubDate: 2017-09-27 DOI: 10.1007/s00012-017-0455-y

Authors:Gábor Czédli Abstract: Abstract By a 1991 result of R. Freese, G. Grätzer, and E. T. Schmidt, every complete lattice A is isomorphic to the lattice Com(K) of complete congruences of a strongly atomic, 3-distributive, complete modular lattice K. In 2002, Grätzer and Schmidt improved 3-distributivity to 2-distributivity. Here, we represent morphisms between two complete lattices with complete lattice congruences in three ways. Namely, for \({i \in \{1, 2, 3\},}\) let \({A_i}\) and \({A^{\prime}_{i}}\) be arbitrary complete lattices and let \({f_i}\) : \({A_{i} \rightarrow A^{\prime}_{i}}\) be maps such that (i) \({f_1}\) is \({(\bigvee, 0)}\) -preserving and 0-separating, (ii) \({f_2}\) is \({(\bigwedge, 0, 1)}\) -preserving, and (iii) \({f_3}\) is \({(\bigvee, 0)}\) -preserving. We prove that for \({i \in \{1, 2, 3\},}\) there exist strongly atomic, 2-distributive, complete modular lattices \({K_i}\) and \({K^{\prime}_{i}}\) such that \({A_{i} \cong {\rm Com}({K_{i}}), A^{\prime}_{i} \cong {\rm Com}({K^{\prime}_{i}}),}\) and, in addition, (i) \({K_1}\) is a principal ideal of \({{K^{\prime}_{1}}}\) and \({f_1}\) is represented by complete congruence extension, (ii) \({K^{\prime}_{2}}\) is a sublattice of \({K_2}\) and \({f_2}\) is represented by restriction, and (iii) \({f_3}\) is represented as the composite of a map naturally induced by a complete lattice homomorphism from \({K_3}\) to \({K^{\prime}_{3}}\) and the complete congruence generation in \({K^{\prime}_{3}}\) . Also, our approach yields a relatively easy construction that proves the above-mentioned 2002 result of Grätzer and Schmidt. PubDate: 2017-09-27 DOI: 10.1007/s00012-017-0457-9

Authors:Dragan Mašulović; Lynn Scow Abstract: Abstract In this paper, we investigate the best known and most important example of a categorical equivalence in algebra, that between the variety of boolean algebras and any variety generated by a single primal algebra. We consider this equivalence in the context of Kechris-Pestov-Todorčević correspondence, a surprising correspondence between model theory, combinatorics and topological dynamics. We show that relevant combinatorial properties (such as the amalgamation property, Ramsey property and ordering property) carry over from a category to an equivalent category. We then use these results to show that the category whose objects are isomorphic copies of finite powers of a primal algebra \({\mathcal{A}}\) together with a particular linear ordering <, and whose morphisms are embeddings, is a Ramsey age (and hence a Fraïssé age). By the Kechris-Pestov-Todorčević correspondence, we then infer that the automorphism group of its Fraïssé limit is extremely amenable. This correspondence also enables us to compute the universal minimal flow of the Fraïssé limit of the class \({{\bf V}_{fin} \mathcal{(A)}}\) whose objects are isomorphic copies of finite powers of a primal algebra \({\mathcal{A}}\) and whose morphisms are embeddings. PubDate: 2017-08-09 DOI: 10.1007/s00012-017-0453-0

Authors:Sergio A. Celani; María Esteban Abstract: Abstract Distributive Hilbert algebras with infimum, or DH^-algebras for short, are algebras with implication and conjunction, in which the implication and the conjunction do not necessarily satisfy the residuation law. These algebras do not fall under the scope of the usual duality theory for lattice expansions, precisely because they lack residuation. We propose a new approach, that consists of regarding the conjunction as the additional operation on the underlying implicative structure. In this paper, we introduce a class of spaces, based on compactly-based sober topological spaces. We prove that the category of these spaces and certain relations is dually equivalent to the category of DH^-algebras and \({\wedge}\) -semi-homomorphisms. We show that the restriction of this duality to a wide subcategory of spaces gives us a duality for the category of DH^-algebras and algebraic homomorphisms. This last duality generalizes the one given by the author in 2003 for implicative semilattices. Moreover, we use the duality to give a dual characterization of the main classes of filters for DH^-algebras, namely, (irreducible) meet filters, (irreducible) implicative filters and absorbent filters. PubDate: 2017-07-26 DOI: 10.1007/s00012-017-0451-2

Authors:Brett McLean Abstract: Abstract We define antidomain operations for algebras of multiplace partial functions. For all signatures containing composition, the antidomain operations and any subset of intersection, preferential union and fixset, we give finite equational or quasiequational axiomatisations for the representation class. We do the same for the question of representability by injective multiplace partial functions. For all our representation theorems, it is an immediate corollary of our proof that the finite representation property holds for the representation class. We show that for a large set of signatures, the representation classes have equational theories that are coNP-complete. PubDate: 2017-07-19 DOI: 10.1007/s00012-017-0452-1

Authors:Anvar M. Nurakunov Abstract: Abstract An algebraic structure A is said to be finitely subdirectly reducible if A is not finitely subdirectly irreducible. We show that for any signature providing only finitely many relation symbols, the class of finitely subdirectly reducible algebraic structures is closed with respect to the formation of ultraproducts. We provide some corollaries and examples for axiomatizable classes that are closed with respect to the formation of finite subdirect products, in particular, for varieties and quasivarieties. PubDate: 2017-07-19 DOI: 10.1007/s00012-017-0450-3

Authors:Kirby A. Baker Abstract: Abstract This note is an addendum to clarify credit for universal relational systems and their properties. PubDate: 2017-05-24 DOI: 10.1007/s00012-017-0449-9

Authors:T. S. Blyth; H. J. Silva Abstract: Abstract We consider, in the context of an Ockham algebra \({{\mathcal{L} = (L; f)}}\) , the ideals I of L that are kernels of congruences on \({\mathcal{L}}\) . We describe the smallest and the largest congruences having a given kernel ideal, and show that every congruence kernel \({I \neq L}\) is the intersection of the prime ideals P such that \({I \subseteq P}\) , \({P \cap f(I) = \emptyset}\) , and \({f^{2}(I) \subseteq P}\) . The congruence kernels form a complete lattice which in general is not modular. For every non-empty subset X of L, we also describe the smallest congruence kernel to contain X, in terms of which we obtain necessary and sufficient conditions for modularity and distributivity. The case where L is of finite length is highlighted. PubDate: 2017-05-22 DOI: 10.1007/s00012-017-0441-4

Authors:M. Andrew Moshier; Jorge Picado; Aleš Pultr Abstract: Abstract Generalizing the obvious representation of a subspace \({Y \subseteq X}\) as a sublocale in Ω(X) by the congruence \({\{(U, V ) U\cap Y = V \cap Y\}}\) , one obtains the congruence \({\{(a, b) \mathfrak{o}(a) \cap S = \mathfrak{o}(b) \cap S\}}\) , first with sublocales S of a frame L, which (as it is well known) produces back the sublocale S itself, and then with general subsets \({S\subseteq L}\) . The relation of such S with the sublocale produced is studied (the result is not always the sublocale generated by S). Further, we discuss in general the associated adjunctions, in particular that between relations on L and subsets of L and view the aforementioned phenomena in this perspective. PubDate: 2017-05-22 DOI: 10.1007/s00012-017-0446-z