Authors:Gábor Czédli Pages: 251 - 289 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: 2017-11-01 DOI: 10.1007/s00012-017-0457-9 Issue No:Vol. 78, No. 3 (2017)

Authors:Miguel A. Campercholi; James G. Raftery Pages: 407 - 425 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: 2017-11-01 DOI: 10.1007/s00012-017-0455-y Issue No:Vol. 78, No. 3 (2017)

Authors:Antonio Boccuto; Antonio Di Nola; Gaetano Vitale Abstract: Affine representations for archimedean \({\ell}\) -groups and semisimple MV-algebras via embedding theorems are presented; they are simple to work with but powerful enough to express significant properties of our studied objects. Indeed, we focus on the space of particular homomorphisms between an archimedean \({\ell}\) -group (a semisimple MV-algebra, respectively) and a vector lattice (a Riesz MV-algebra, respectively), i.e., the set of the generalized states, providing a general framework. PubDate: 2017-11-08 DOI: 10.1007/s00012-017-0477-5

Authors:Walter Taylor Abstract: This paper is a brief study of the equations compatible with the (topological) n-book for all \({n \geq 2}\) . For each n, we exhibit an equation-set (an extension of lattice theory) that is identically satisfiable on the \({(n + 1)}\) -book, but not on the n-book. PubDate: 2017-11-07 DOI: 10.1007/s00012-017-0478-4

Authors:Keith A. Kearnes; Ágnes Szendrei; Ross Willard Abstract: This paper is motivated by a practical question: given a finite algebra A in a finite language, how can we best program a computer to decide whether the variety generated by A has a difference term, and how hard is it to find the difference term' To help address this question, we produce a simple Maltsev condition which characterizes difference terms in the class of locally finite varieties. We do the same for weak difference terms. PubDate: 2017-11-06 DOI: 10.1007/s00012-017-0475-7

Authors:Tim Penttila Abstract: In 1995, Pálfy and Szabó stated the theorem that a projective space satisfies the six-cross theorem if and only if it is Desarguesian. A mistake in their proof is corrected. Moreover, for projective planes, the three-cross theorem of Pálfy and Szabó is identified as the Reidemeister condition. PubDate: 2017-11-06 DOI: 10.1007/s00012-017-0473-9

Authors:Keith A. Kearnes; Ágnes Szendrei Abstract: We show that an idempotent variety has a d-dimensional cube term if and only if its free algebra on two generators has no d-ary compatible cross. We employ Hall’s Marriage Theorem to show that an idempotent variety \({\mathcal{V}}\) of finite signature whose fundamental operations have arities n 1, . . . , n k, has a d-dimensional cube term for some d if and only if it has one of dimension \({1 + \sum_{i=1}^{k} (n_{i} - 1)}\) . This upper bound on the dimension of a minimal-dimension cube term for \({\mathcal{V}}\) is shown to be sharp. We show that a pure cyclic term variety has a cube term if and only if it contains no 2- element semilattice. We prove that the Maltsev condition “existence of a cube term” is join prime in the lattice of idempotent Maltsev conditions. PubDate: 2017-11-06 DOI: 10.1007/s00012-017-0476-6

Authors:Diego N. Castaño Abstract: In this article, we explore in some detail the free and weakly projective objects of the variety of Łukasiewicz implication algebras (the implicative subreducts of MV-algebras). We review the two already known descriptions of finitely generated free algebras, giving new insights into their structure and their connection, as well as providing new proofs of the characterizations. We give a representation theorem for weakly projective algebras as algebras of certain McNaughton functions restricted to rational polyhedra and prove that finitely generated weakly projective algebras coincide with finitely presented ones. We also prove that finite chains are the only totally ordered weakly projective examples in this variety. PubDate: 2017-11-06 DOI: 10.1007/s00012-017-0474-8

Authors:Jingjing Ma; Warren Wm. McGovern Abstract: Fuchs called a partially-ordered integral domain, say D, division closed if it has the property that whenever a > 0 and ab > 0, then b > 0. He showed that if D is a lattice-ordered division closed field, then D is totally ordered. In fact, it is known that for a lattice-ordered division ring, the following three conditions are equivalent: a) squares are positive, b) the order is total, and c) the ring is division closed. In the present article, our aim is to study \({\ell}\) -rings that possibly possess zerodivisors and focus on a natural generalization of the property of being division closed, what we call regular division closed. Our investigations lead us to the concept of a positive separating element in an \({\ell}\) -ring, which is related to the well-known concept of a positive d-element. PubDate: 2017-10-23 DOI: 10.1007/s00012-017-0467-7

Authors:Marcel Erné; Jorge Picado Abstract: A classical tensor product \({A \otimes B}\) of complete lattices A and B, consisting of all down-sets in \({A \times B}\) that are join-closed in either coordinate, is isomorphic to the complete lattice Gal(A,B) of Galois maps from A to B, turning arbitrary joins into meets. We introduce more general kinds of tensor products for closure spaces and for posets. They have the expected universal property for bimorphisms (separately continuous maps or maps preserving restricted joins in the two components) into complete lattices. The appropriate ingredient for quantale constructions is here distributivity at the bottom, a generalization of pseudocomplementedness. We show that the truncated tensor product of a complete lattice B with itself becomes a quantale with the closure of the relation product as multiplication iff B is pseudocomplemented, and that the tensor product has a unit element iff B is atomistic. The pseudocomplemented complete lattices form a semicategory in which the hom-set between two objects is their tensor product. The largest subcategory of that semicategory has as objects the atomic boolean complete lattices, which is equivalent to the category of sets and relations. More general results are obtained for closure spaces and posets. PubDate: 2017-10-22 DOI: 10.1007/s00012-017-0472-x

Authors:Erhard Aichinger Abstract: We consider clones on countable sets. If such a clone has quasigroup operations, is locally closed and countable, then there is a function \({f : \mathbb{N} \rightarrow \mathbb{N}}\) such that the n-ary part of C is equal to the n-ary part of Pol \({{\rm Inv}^{[f(n)]} C}\) , where \({{\rm Inv}^{[f(n)]} C}\) denotes the set of f(n)-ary invariant relations of C. PubDate: 2017-10-22 DOI: 10.1007/s00012-017-0465-9

Authors:Patrick Cégielski; Serge Grigorieff; Irène Guessarian Abstract: A function on an algebra is congruence preserving if for any congruence, it maps congruent elements to congruent elements. We show that on a free monoid generated by at least three letters, a function from the free monoid into itself is congruence preserving if and only if it is of the form \({x \mapsto w_{0}xw_{1} \cdots w_{n-1}xw_n }\) for some finite sequence of words \({w_0,\ldots ,w_n}\) . We generalize this result to functions of arbitrary arity. This shows that a free monoid with at least three generators is a (noncommutative) affine complete algebra. As far as we know, it is the first (nontrivial) case of a noncommutative affine complete algebra. PubDate: 2017-10-20 DOI: 10.1007/s00012-017-0464-x

Authors:G. Grätzer 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: 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: 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: 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: 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: 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: 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