Authors:José Luis Castiglioni; Sergio Arturo Celani; Hernán Javier San Martín Pages: 375 - 393 Abstract: Inspired by an old construction due to J. Kalman that relates distributive lattices and centered Kleene algebras, in this paper we study an equivalence for certain categories whose objects are algebras with implication \({(H, \bigwedge, \bigvee, \rightarrow, 0,1)}\) which satisfy the following property for every \({a,b,c\, \in\, H}\) : if \({a \leq b \rightarrow c}\) , then \({a \bigwedge b \leq c}\) . PubDate: 2017-07-01 DOI: 10.1007/s00012-017-0433-4 Issue No:Vol. 77, No. 4 (2017)

Authors:Miao Miao Ren; Xian Zhong Zhao; Ai Fa Wang Pages: 395 - 408 Abstract: The aim of this paper is to study the varieties of ai-semirings satisfying \({x^{3}\approx x}\) . It is shown that the collection of all such varieties forms a distributive lattice of order 179. Also, all of them are finitely based and finitely generated. This generalizes and extends the main results obtained by Ghosh et al., Pastijn and Ren and Zhao. PubDate: 2017-07-01 DOI: 10.1007/s00012-017-0438-z Issue No:Vol. 77, No. 4 (2017)

Authors:Niels Schwartz Pages: 409 - 442 Abstract: A spectral space is localic if it corresponds to a frame under Stone Duality. This class of spaces was introduced by the author (under the name ’locales’) as the topological version of the classical frame theoretic notion of locales, see Johnstone and also Picado and Pultr). The appropriate class of subspaces of a localic space are the localic subspaces. These are, in particular, spectral subspaces. The following main questions are studied (and answered): Given a spectral subspace of a localic space, how can one recognize whether the subspace is even localic? How can one construct all localic subspaces from particularly simple ones? The set of localic subspaces and the set of spectral subspaces are both inverse frames. The set of localic subspaces is known to be the image of an inverse nucleus on the inverse frame of spectral subspaces. How can the inverse nucleus be described explicitly? Are there any special properties distinguishing this particular inverse nucleus from all others? Colimits of spectral spaces and localic spaces are needed as a tool for the comparison of spectral subspaces and localic subspaces. PubDate: 2017-07-01 DOI: 10.1007/s00012-017-0436-1 Issue No:Vol. 77, No. 4 (2017)

Authors:Gábor Czédli Pages: 443 - 498 Abstract: In 2009, G. Grätzer and E. Knapp proved that every planar semimodular lattice has a rectangular extension. We prove that, under reasonable additional conditions, this extension is unique. This theorem naturally leads to a hierarchy of special diagrams of planar semimodular lattices. These diagrams are unique in a strong sense; we also explore many of their additional properties. We demonstrate the power of our new classes of diagrams in two ways. First, we prove a simplified version of our earlier Trajectory Coloring Theorem, which describes the inclusion con \({(\mathfrak{p}) \supseteq}\) con \({(\mathfrak{q})}\) for prime intervals \({\mathfrak{p}}\) and \({\mathfrak{q}}\) in slim rectangular lattices. Second, we prove G. Grätzer’s Swing Lemma for the same class of lattices, which describes the same inclusion more simply. PubDate: 2017-07-01 DOI: 10.1007/s00012-017-0437-0 Issue No:Vol. 77, No. 4 (2017)

Authors:Greg Oman 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: 2017-05-30 DOI: 10.1007/s00012-017-0448-x

Authors:Kirby A. Baker 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:Přemysl Jedlička; Agata Pilitowska; Anna Zamojska-Dzienio Abstract: This paper gives the construction of free medial quandles as well as free n-symmetric medial quandles and free m-reductive medial quandles. PubDate: 2017-05-23 DOI: 10.1007/s00012-017-0443-2

Authors:Anatolij Dvurečenskij; Omid Zahiri Abstract: We study conditions when a certain type of the Riesz Decomposition Property (RDP for short) holds in the lexicographic product of two po-groups. Defining two important properties of po-groups, we extend known situations showing that the lexicographic product satisfies RDP or even \({{\rm RDP}_1}\) , a stronger type of RDP. We recall that a very strong type of RDP, \({{\rm RDP}_2}\) , entails that the group is lattice ordered. RDP's of the lexicographic products are important for the study of lexicographic pseudo effect algebras, or perfect types of pseudo MV-algebras and pseudo effect algebras, where infinitesimal elements play an important role both for algebras as well as for the first order logic of valid but not provable formulas. PubDate: 2017-05-23 DOI: 10.1007/s00012-017-0447-y

Authors:Graham Manuell 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: 2017-05-22 DOI: 10.1007/s00012-017-0439-y

Authors:T. S. Blyth; H. J. Silva 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: 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

Authors:Karim Boulabiar; Chiheb El Adeb Abstract: Recently, Ball defined a truncated \({\ell}\) -group to be an \({\ell}\) -group G along with a truncation. We constructively prove that if G is a truncated \({\ell}\) -group, then the direct sum \({G \oplus \mathbb{Q}}\) is equipped with a structure of an \({\ell}\) -group with weak unit the rational number 1. As a simple consequence, we get a description of the truncated \({\ell}\) - group obtained by Ball via representation theory. On the other hand, we derive some characterizations of truncation morphisms as defined by Ball himself. In particular, we show that the group homomorphism \({f : G \rightarrow H}\) is a truncation morphism if and only its natural extension \({f^*}\) from \({G \oplus \mathbb{Q}}\) into \({H \oplus \mathbb{Q}}\) is an \({\ell}\) -homomorphism. PubDate: 2017-05-22 DOI: 10.1007/s00012-017-0444-1

Authors:Edmond W. H. Lee 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: 2017-05-22 DOI: 10.1007/s00012-017-0442-3

Authors:Youssef Azouzi; Mohamed Amine Ben Amor Abstract: Let B be an Archimedean reduced f-ring. A positive element \({\omega}\) in B is said to satisfy the property \({(\ast)}\) if for every f-ring A with identity e and every \({\ell}\) -group homomorphism \({\gamma : A \rightarrow B}\) with \({\gamma(e) = \omega}\) , there exists a unique \({\ell}\) -ring homomorphism \({\rho: B \rightarrow B}\) such that \({\gamma = \omega \rho}\) and \({\rho(e)^{\perp \perp} = \omega^{\perp \perp}}\) . Boulabiar and Hager proved that any (positive) von Neumann regular element in B satisfies the property \({(\ast)}\) and proved that the converse holds in the C(X)-case. In this regard, they asked about this converse in the general case. Our main purpose in this note is to prove, via a counter-example, that the converse in question fails in general. In addition, we shall take the opportunity to extend the direct result obtained by Boulabiar and Hager, and to get the C(X)-case we were talking about in an easier way. PubDate: 2017-05-22 DOI: 10.1007/s00012-017-0445-0

Authors:Lawrence Peter Belluce; Antonio Di Nola; Giacomo Lenzi Abstract: Given an MV-algebra A, with its natural partial ordering, we consider in A the intervals of the form [0, a], where \({a \in A}\) . These intervals have a natural structure of MV-algebras and will be called the relative subalgebras of A (in analogy with Boolean algebras). We investigate various properties of relative subalgebras and their relations with the original MV-algebra. PubDate: 2017-02-09 DOI: 10.1007/s00012-017-0435-2

Authors:Anatolii V. Zhuchok Abstract: In this paper, we consider doppelsemigroups, which are sets with two binary associative operations satisfying additional axioms. Commutative dimonoids in the sense of Loday are examples of doppelsemigroups and two interassociative semigroups give rise to a doppelsemigroup. The main result of this paper is the construction of the free product of doppelsemigroups. We also construct the free doppelsemigroup, the free commutative doppelsemigroup, the free n-nilpotent doppelsemigroup, and characterize the least commutative congruence and the least n-nilpotent congruence on a free doppelsemigroup. PubDate: 2017-02-09 DOI: 10.1007/s00012-017-0431-6

Authors:Victoria Gould; Tim Stokes Abstract: Constellations are partial algebras that are one-sided generalisations of categories. Indeed, we show that a category is exactly a constellation that also satisfies the left-right dual axioms. Constellations have previously appeared in the context of inductive constellations: the category of inductive constellations is known to be isomorphic to the category of left restriction semigroups. Here we consider constellations in full generality, giving many examples. We characterise those small constellations that are isomorphic to constellations of partial functions. We examine in detail the relationship between constellations and categories. In particular, we characterise those constellations that arise as (sub-)reducts of categories. We demonstrate that the notion of substructure can be captured within constellations but not within categories. We show that every constellation P gives rise to a category \({\mathcal{C}(P)}\) , its canonical extension, in a simplest possible way, and that P is a quotient of \({\mathcal{C}(P)}\) in a natural sense. We also show that many of the most common concrete categories may be constructed from simpler quotient constellations using this construction. We characterise the canonical congruences \({\delta}\) on a given category \({K}\) (those for which \({K \cong \mathcal{C}(K/\delta))}\) , and show that the category of constellations is equivalent to the category of \({\delta}\) -categories, that is, categories equipped with distinguished canonical congruence \({\delta}\) . The main observation of this paper is that category theory as it applies to the familiar concrete categories of modern mathematics (which come equipped with natural notions of substructures and indeed are \({\delta}\) -categories) may be subsumed by constellation theory. PubDate: 2017-02-09 DOI: 10.1007/s00012-017-0432-5

Authors:Manuel Bodirsky; Friedrich Martin Schneider Abstract: A topological monoid is isomorphic to an endomorphism monoid of a countable structure if and only if it is separable and has a compatible complete ultrametric such that composition from the left is non-expansive. We also give a topological characterisation of those topological monoids that are isomorphic to endomorphism monoids of countable \({\omega}\) -categorical structures. Finally, we present analogous characterisations for polymorphism clones of countable structures and for polymorphism clones of countable \({\omega}\) -categorical structures. PubDate: 2017-02-06 DOI: 10.1007/s00012-017-0427-2

Authors:Jeannine J. M. Gabriëls; Stephen M. Gagola; Mirko Navara Abstract: We collect, correct, and extend results on the properties of the Sasaki projection in orthomodular lattices. We bring arguments as to why this operation can extend tools for simplification of formulas and automated computing. PubDate: 2017-02-01 DOI: 10.1007/s00012-017-0428-1

Authors:Nick Bezhanishvili; Nick Galatos; Luca Spada Abstract: Canonical formulas are a powerful tool for studying intuitionistic and modal logics. Indeed, they provide a uniform and semantic way of axiomatising all extensions of intuitionistic logic and all modal logics above K4. Although the method originally hinged on the relational semantics of those logics, recently it has been completely recast in algebraic terms. In this new perspective, canonical formulas are built from a finite subdirectly irreducible algebra by describing completely the behaviour of some operations and only partially the behaviour of some others. In this paper, we export the machinery of canonical formulas to substructural logics by introducing canonical formulas for k-potent, commutative, integral, residuated lattices (k-CIRL). We show that any subvariety of k-CIRL is axiomatised by canonical formulas. The paper ends with some applications and examples. PubDate: 2017-02-01 DOI: 10.1007/s00012-017-0430-7