Authors:Miao Miao Ren; Xian Zhong Zhao; Ai Fa Wang 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: 2017-03-31 DOI: 10.1007/s00012-017-0438-z

Authors:Gábor Czédli 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: 2017-03-31 DOI: 10.1007/s00012-017-0437-0

Authors:Niels Schwartz 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: 2017-03-31 DOI: 10.1007/s00012-017-0436-1

Authors:Mariana Badano; Diego J. Vaggione Abstract: We study four types of equational definability of factor congruences in varieties with \({\vec{0}}\) and \({\vec{1}}\) . The paper completes the work of a previous paper on left equational definability of factor congruences. PubDate: 2017-02-25 DOI: 10.1007/s00012-017-0434-3

Authors:José Luis Castiglioni; Sergio Arturo Celani; Hernán Javier San Martín 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: 2017-02-10 DOI: 10.1007/s00012-017-0433-4

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:Miklós Dormán Abstract: In this paper, we investigate transformation monoids that are built up from inverse transformation monoids constructed from finite lattices by adding all the unary constant transformations. We give a complete description for the corresponding monoidal intervals in the clone lattice. PubDate: 2017-02-03 DOI: 10.1007/s00012-017-0425-4

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

Authors:Juan M. Cornejo; Hanamantagouda P. Sankappanavar Abstract: In a paper published in 2012, the second author extended the well-known fact that Boolean algebras can be defined using only implication and a constant, to De Morgan algebras—this result led him to introduce, and investigate (in the same paper), the variety \({\mathcal{I}}\) of algebras, there called implication zroupoids (I-zroupoids) and here called implicator groupoids ( \({\mathcal{I}}\) -groupoids), that generalize De Morgan algebras. The present paper is a continuation of the paper mentioned above and is devoted to investigating the structure of the lattice of subvarieties of \({\mathcal{I}}\) , and also to making further contributions to the theory of implicator groupoids. Several new subvarieties of \({\mathcal{I}}\) are introduced and their relationship with each other, and with the subvarieties of \({\mathcal{I}}\) which were already investigated in the paper mentioned above, are explored. PubDate: 2017-02-01 DOI: 10.1007/s00012-017-0429-0

Authors:Dmitriy N. Zhuk Abstract: In the paper, we introduce a notion of a key relation, which is similar to the notion of a critical relation introduced by Keith A. Kearnes and Ágnes Szendrei. All clones on finite sets can be defined by only key relations. In addition, there is a nice description of all key relations on 2 elements. These are exactly the relations that can be defined as a disjunction of linear equations. In the paper, we show that in general, key relations do not have such a nice description. Nevertheless, we obtain a nice characterization of all key relations preserved by a weak near-unanimity function. This characterization is presented in the paper. PubDate: 2017-01-31 DOI: 10.1007/s00012-017-0426-3

Authors:Mojgan Mahmoudi; Halimeh Moghbeli; Konrad Pióro Abstract: Directed complete partially ordered sets (dcpos, for short) play an important role in domain theory. The aim of this paper is to characterise natural congruences of dcpos. We also show that the kernels of dcpo maps, that is, directed join-preserving maps between dcpos are not necessarily natural dcpo congruences. Then we characterise dcpo maps whose kernels are natural dcpo congruences. Finally, we prove the Decomposition and Isomorphism Theorems for dcpo maps. PubDate: 2017-01-18 DOI: 10.1007/s00012-017-0424-5

Authors:Zurab Janelidze; Thomas Weighill Abstract: Normal categories are pointed categorical counterparts of 0-regular varieties, i.e., varieties where each congruence is uniquely determined by the equivalence class of a fixed constant 0. In this paper, we give a new axiomatic approach to normal categories, which uses self-dual axioms on a functor defined using subobjects of objects in the category. We also show that a similar approach can be developed for 0-regular varieties, if we replace subobjects with subsets of algebras containing 0. PubDate: 2017-01-17 DOI: 10.1007/s00012-017-0422-7

Authors:Jean B. Nganou Abstract: Characterizations of compact Hausdorff topological MV-algebras, Stone MV-algebras, and MV-algebras that are isomorphic to their profinite completions are established. It is proved that compact Hausdorff topological MV-algebras are products (both topological and algebraic) of copies [0, 1] with the interval topology and finite Łukasiewicz chains with the discrete topology. Going one step further, we also prove that Stone MV-algebras are products (both topological and algebraic) of finite Łukasiewicz chains with the discrete topology. Finally, it is proved that an MV-algebra is isomorphic to its profinite completion if and only if it is profinite and each of its maximal ideals of finite rank is principal. PubDate: 2017-01-17 DOI: 10.1007/s00012-016-0421-0

Authors:C. Jayaram Abstract: In this paper, we prove that an indecomposable M-lattice is either a principal element domain or a special principal element lattice. Next, we introduce weak complemented elements and characterize reduced M-lattices in terms of weak complemented elements. We also study weak invertible elements and locally weak invertible elements in C-lattices and characterize reduced Prüfer lattices, WI-lattices, reduced almost principal element lattices, and reduced principal element lattices in terms of locally weak invertible elements. PubDate: 2017-01-17 DOI: 10.1007/s00012-017-0423-6

Authors:Miguel Couceiro; Lucien Haddad; Karsten Schölzel; Tamás Waldhauser Abstract: D. Lau raised the problem of determining the cardinality of the set of all partial clones of Boolean functions whose total part is a given Boolean clone. The key step in the solution of this problem, which was obtained recently by the authors, was to show that the sublattice of strong partial clones on \(\{0, 1\}\) that contain all total functions preserving the relation \({\rho_{0,2} = \{(0, 0), (0, 1), (1, 0)\}}\) is of continuum cardinality. In this paper, we represent relations derived from \({\rho_{0,2}}\) in terms of graphs, and we define a suitable closure operator on graphs such that the lattice of closed sets of graphs is isomorphic to the dual of this uncountable sublattice of strong partial clones. With the help of this duality, we provide a rough description of the structure of this lattice, and we also obtain a new proof for its uncountability. PubDate: 2017-01-06 DOI: 10.1007/s00012-016-0418-8

Authors:Gábor Czédli Abstract: For a lattice L with 0 and 1, let Princ(L) denote the set of principal congruences of L. Ordered by set inclusion, it is a bounded ordered set. In 2013, G. Grätzer proved that every bounded ordered set is representable as Princ(L); in fact, he constructed L as a lattice of length 5. For {0, 1}-sublattices \({A \subseteq B}\) of L, congruence generation defines a natural map Princ(A) \({\longrightarrow}\) Princ(B). In this way, every family of {0, 1}-sublattices of L yields a small category of bounded ordered sets as objects and certain 0-separating {0, 1}-preserving monotone maps as morphisms such that every hom-set consists of at most one morphism. We prove the converse: every small category of bounded ordered sets with these properties is representable by principal congruences of selfdual lattices of length 5 in the above sense. As a corollary, we can construct a selfdual lattice L in G. Grätzer's above-mentioned result. PubDate: 2017-01-03 DOI: 10.1007/s00012-016-0419-7