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:Libor Barto; Jakub Bulín Pages: 3 - 18 Abstract: Abstract We prove that for finite, finitely related algebras, the concepts of an absorbing subuniverse and a J´onsson absorbing subuniverse coincide. Consequently, it is decidable whether a given subset is an absorbing subuniverse of the polymorphism algebra of a given relational structure. PubDate: 2017-09-01 DOI: 10.1007/s00012-017-0440-5 Issue No:Vol. 78, No. 1 (2017)

Authors:Mariana Badano; Diego J. Vaggione Pages: 19 - 42 Abstract: 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: 2017-09-01 DOI: 10.1007/s00012-017-0434-3 Issue No:Vol. 78, No. 1 (2017)

Authors:Karim Boulabiar; Chiheb El Adeb Pages: 93 - 104 Abstract: 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: 2017-09-01 DOI: 10.1007/s00012-017-0444-1 Issue No:Vol. 78, No. 1 (2017)

Authors:Youssef Azouzi; Mohamed Amine Ben Amor Pages: 119 - 124 Abstract: 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: 2017-09-01 DOI: 10.1007/s00012-017-0445-0 Issue No:Vol. 78, No. 1 (2017)

Authors:José Luis Castiglioni; Sergio Arturo Celani; Hernán Javier San Martín Pages: 375 - 393 Abstract: 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: 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: 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: 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: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:Přemysl Jedlička; Agata Pilitowska; Anna Zamojska-Dzienio Abstract: 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: 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: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