Authors:Laurent De Rudder; Georges Hansoul Abstract: It is well known that the category of compact Hausdorff spaces is dually equivalent to the category of commutative \(C^\star \) -algebras. More generally, this duality can be seen as a part of a square of dualities and equivalences between compact Hausdorff spaces, \(C^\star \) -algebras, compact regular frames and de Vries algebras. Three of these equivalences have been extended to equivalences between compact pospaces, stably compact frames and proximity frames, the fourth part of what will be a second square being lacking. We propose the category of bounded Archimedean \(\ell \) -semi-algebras to complete the second square of equivalences and to extend the category of \(C^\star \) -algebras. PubDate: 2018-05-17 DOI: 10.1007/s00012-018-0519-7 Issue No:Vol. 79, No. 2 (2018)

Authors:J. B. Nation; Joy Nishida Abstract: A stronger version of a known property is shown to hold for the natural equaclosure operator on subquasivariety lattices. PubDate: 2018-05-14 DOI: 10.1007/s00012-018-0518-8 Issue No:Vol. 79, No. 2 (2018)

Authors:George F. McNulty; Ross Willard Abstract: We provide several conditions that, among locally finite varieties, characterize congruence meet-semidistributivity and we use these conditions to give a new proof of a finite basis theorem published by Baker, McNulty, and Wang in 2004. This finite basis theorem extends Willard’s Finite Basis Theorem. PubDate: 2018-05-14 DOI: 10.1007/s00012-018-0524-x Issue No:Vol. 79, No. 2 (2018)

Authors:Christian Pech; Maja Pech Abstract: In this paper, motivated by classical results by Sierpiński, Arnold and Kolmogorov, we derive sufficient conditions for polymorphism clones of homogeneous structures to have a generating set of bounded arity. We use our findings in order to describe a class of homogeneous structures whose polymorphism clones have a finite Sierpiński rank, uncountable cofinality, and the Bergman property. PubDate: 2018-05-14 DOI: 10.1007/s00012-018-0527-7 Issue No:Vol. 79, No. 2 (2018)

Authors:Igor Dolinka Abstract: Headlining the Topical Collection dedicated to The 5th Novi Sad Algebraic Conference (NSAC 2017), we provide a brief report of the conference along with some of its history and background. PubDate: 2018-05-09 DOI: 10.1007/s00012-018-0528-6 Issue No:Vol. 79, No. 2 (2018)

Authors:Bertalan Bodor; Peter J. Cameron; Csaba Szabó Abstract: It is shown that the countably infinite dimensional pointed vector space (the vector space equipped with a constant) over a finite field has infinitely many first order definable reducts. This implies that the countable homogeneous Boolean-algebra has infinitely many reducts. PubDate: 2018-05-09 DOI: 10.1007/s00012-018-0526-8 Issue No:Vol. 79, No. 2 (2018)

Authors:Guillermo Badia; João Marcos Abstract: We provide universal algebraic characterizations (in the sense of not involving any “logical notion”) of some elementary classes of structures whose definitions involve universal d-Horn sentences and universally closed disjunctions of atomic formulas. These include, in particular, the classes of fields, of non-trivial rings, and of directed graphs without loops where every two elements are adjacent. The classical example of this kind of characterization result is the HSP theorem, but there are myriad other examples (e.g., the characterization of elementary classes using isomorphic images, ultraproducts and ultrapowers due to Keisler and Shelah). PubDate: 2018-05-07 DOI: 10.1007/s00012-018-0522-z Issue No:Vol. 79, No. 2 (2018)

Authors:Gábor Czédli; George Grätzer; Harry Lakser Abstract: The Swing Lemma, proved by G. Grätzer in 2015, describes how a congruence spreads from a prime interval to another in a slim (having no \(\mathsf {M}_{3}\) sublattice), planar, semimodular lattice. We generalize the Swing Lemma to planar semimodular lattices. PubDate: 2018-04-30 DOI: 10.1007/s00012-018-0483-2 Issue No:Vol. 79, No. 2 (2018)

Authors:Joe Cyr Abstract: It is known that for finite algebras, solvable implies hereditarily absorption free. We present an example which shows that this implication does not hold for infinite algebras. This example is also quasi-affine, contradicting an earlier statement that quasi-affine algebras are hereditarily absorption free. PubDate: 2018-04-24 DOI: 10.1007/s00012-018-0521-0 Issue No:Vol. 79, No. 2 (2018)

Authors:Richard N. Ball; Aleš Pultr Abstract: We show that every frame can be essentially embedded in a Boolean frame, and that this embedding is the maximal essential extension of the frame in the sense that it factors uniquely through any other essential extension. This extension can be realized as the embedding \(L \rightarrow \mathcal {N}(L) \rightarrow \mathcal {B}\mathcal {N}(L)\) , where \(L \rightarrow \mathcal {N}(L)\) is the familiar embedding of L into its congruence frame \(\mathcal {N}(L)\) , and \(\mathcal {N}(L) \rightarrow \mathcal {B}\mathcal {N}(L)\) is the Booleanization of \(\mathcal {N}(L)\) . Finally, we show that for subfit frames the extension can also be realized as the embedding \(L \rightarrow {{\mathrm{S}}}_\mathfrak {c}(L)\) of L into its complete Boolean algebra \({{\mathrm{S}}}_\mathfrak {c}(L)\) of sublocales which are joins of closed sublocales. PubDate: 2018-04-20 DOI: 10.1007/s00012-018-0508-x Issue No:Vol. 79, No. 2 (2018)

Authors:Sergey V. Gusev Abstract: We completely classify all neutral and costandard elements in the lattice \(\mathbb {MON}\) of all monoid varieties. Further, we prove that an arbitrary upper-modular element of \(\mathbb {MON}\) except the variety of all monoids is either a completely regular or a commutative variety. Finally, we verify that all commutative varieties of monoids are codistributive elements of \(\mathbb {MON}\) . Thus, the problems of describing codistributive or upper-modular elements of \(\mathbb {MON}\) are completely reduced to the completely regular case. PubDate: 2018-04-20 DOI: 10.1007/s00012-018-0513-0 Issue No:Vol. 79, No. 2 (2018)

Authors:Richard L. Kramer; Roger D. Maddux Abstract: An error in a proof of a correct theorem in the classic paper, Boolean Algebras with Operators, Part I, by Jónsson and Tarski is discussed. PubDate: 2018-04-20 DOI: 10.1007/s00012-018-0520-1 Issue No:Vol. 79, No. 2 (2018)

Authors:Niovi Kehayopulu Abstract: We characterize the ordered semigroups that are archimedean and contain an intra-regular element, showing that they are exactly nil extensions of simple ordered semigroups, also the ordered semigroups that are both archimedean and \(\pi \) -semisimple or both archimedean and \(\pi \) -quasi-semisimple. PubDate: 2018-04-20 DOI: 10.1007/s00012-018-0502-3 Issue No:Vol. 79, No. 2 (2018)

Authors:Michał M. Stronkowski Abstract: We present a scheme for providing axiomatizations of universal classes. We use infinitary sentences there. New proofs of Birkhoff’s \(\mathsf {HSP}\) -theorem and Mal’cev’s \(\mathsf {SPP_U}\) -theorem are derived. In total, we present 75 facts of this sort. PubDate: 2018-04-20 DOI: 10.1007/s00012-018-0507-y Issue No:Vol. 79, No. 2 (2018)

Authors:Gergő Gyenizse; Miklós Maróti Abstract: The set \({{\mathrm{Quo}}}(\mathbf {A})\) of compatible quasiorders (reflexive and transitive relations) of an algebra \(\mathbf {A}\) forms a lattice under inclusion, and the lattice \({{\mathrm{Con}}}(\mathbf {A})\) of congruences of \(\mathbf {A}\) is a sublattice of \({{\mathrm{Quo}}}(\mathbf {A})\) . We study how the shape of congruence lattices of algebras in a variety determine the shape of quasiorder lattices in the variety. In particular, we prove that a locally finite variety is congruence distributive [modular] if and only if it is quasiorder distributive [modular]. We show that the same property does not hold for meet semi-distributivity. From tame congruence theory we know that locally finite congruence meet semi-distributive varieties are characterized by having no sublattice of congruence lattices isomorphic to the lattice \(\mathbf {M}_3\) . We prove that the same holds for quasiorder lattices of finite algebras in arbitrary congruence meet semi-distributive varieties, but does not hold for quasiorder lattices of infinite algebras even in the variety of semilattices. PubDate: 2018-04-20 DOI: 10.1007/s00012-018-0512-1 Issue No:Vol. 79, No. 2 (2018)

Authors:Ali Reza Olfati Abstract: Let X be a zero-dimensional space and Y be a Tychonoff space. We show that every non-zero ring homomorphism \(\Phi :C(X,\mathbb {Z})\rightarrow C(Y)\) can be induced by a continuous function \(\pi :Y\rightarrow \upsilon _0X.\) Using this, it turns out that the kernel of such homomorphisms is equal to the intersection of some family of minimal prime ideals in \({{\mathrm{MinMax}}}\left( C(X,\mathbb {Z})\right) .\) As a consequence, we are able to obtain the fact that the factor ring \(\frac{C(X,\mathbb {Z})}{C_F(X,\mathbb {Z})}\) is a subring of some ring of continuous functions if and only if each infinite subset of isolated points of X has a limit point in \(\upsilon _0X.\) This implies that for an arbitrary infinite set X, the factor ring \(\frac{\prod _{_{x\in X}}\mathbb {Z}_{_{x}}}{\oplus _{_{x\in X}}\mathbb {Z}_{_{x}}}\) is not embedded in any ring of continuous functions. The classical ring of quotients of the factor ring \(\frac{C(X,\mathbb {Z})}{C_F(X,\mathbb {Z})}\) is fully characterized. Finally, it is shown that the factor ring \(\frac{C(X,\mathbb {Z})}{C_F(X,\mathbb {Z})}\) is an I-ring if and only if each infinite subset of isolated points on X has a limit point in \(\upsilon _0X\) and \(\upsilon _0X{\setminus }\mathbb {I}(X)\) is an extremally disconnected \(C_{\mathbb {Z}}\) -subspace of \(\upsilon _0X,\) where \(\mathbb {I}(X)\) is the set of all isolated points of X. PubDate: 2018-04-20 DOI: 10.1007/s00012-018-0509-9 Issue No:Vol. 79, No. 2 (2018)

Authors:John Harding; Carol Walker; Elbert Walker Abstract: For L a complete lattice L and \(\mathfrak {X}=(X,(R_i)_I)\) a relational structure, we introduce the convolution algebra \(L^{\mathfrak {X}}\) . This algebra consists of the lattice \(L^X\) equipped with an additional \(n_i\) -ary operation \(f_i\) for each \(n_i+1\) -ary relation \(R_i\) of \(\mathfrak {X}\) . For \(\alpha _1,\ldots ,\alpha _{n_i}\in L^X\) and \(x\in X\) we set \(f_i(\alpha _1,\ldots ,\alpha _{n_i})(x)=\bigvee \{\alpha _1(x_1)\wedge \cdots \wedge \alpha _{n_i}(x_{n_i}):(x_1,\ldots ,x_{n_i},x)\in R_i\}\) . For the 2-element lattice 2, \(2^\mathfrak {X}\) is the reduct of the familiar complex algebra \(\mathfrak {X}^+\) obtained by removing Boolean complementation from the signature. It is shown that this construction is bifunctorial and behaves well with respect to one-one and onto maps and with respect to products. When L is the reduct of a complete Heyting algebra, the operations of \(L^\mathfrak {X}\) are completely additive in each coordinate and \(L^\mathfrak {X}\) is in the variety generated by \(2^\mathfrak {X}\) . Extensions to the construction are made to allow for completely multiplicative operations defined through meets instead of joins, as well as modifications to allow for convolutions of relational structures with partial orderings. Several examples are given. PubDate: 2018-04-20 DOI: 10.1007/s00012-018-0510-3 Issue No:Vol. 79, No. 2 (2018)

Authors:Lucy Ham; Marcel Jackson Abstract: We consider hypergraphs as symmetric relational structures. In this setting, we characterise finite axiomatisability for finitely generated universal Horn classes of loop-free hypergraphs. An Ehrenfeucht–Fraïssé game argument is employed to show that the results continue to hold when restricted to first order definability amongst finite structures. We are also able to show that every interval in the homomorphism order on hypergraphs contains a continuum of universal Horn classes and conclude the article by characterising the intractability of deciding membership in universal Horn classes generated by finite loop-free hypergraphs. PubDate: 2018-04-20 DOI: 10.1007/s00012-018-0515-y Issue No:Vol. 79, No. 2 (2018)

Authors:Christian Pech; Maja Pech Abstract: Every clone of functions comes naturally equipped with a topology, the topology of pointwise convergence. A clone \(\mathfrak {C}\) is said to have automatic homeomorphicity with respect to a class \(\mathcal {K}\) of clones, if every clone isomorphism of \(\mathfrak {C}\) to a member of \(\mathcal {K}\) is already a homeomorphism (with respect to the topology of pointwise convergence). In this paper we study automatic homeomorphicity properties for polymorphism clones of countable homogeneous relational structures. Besides two generic criteria for the automatic homeomorphicity of the polymorphism clones of homogeneous structures we show that the polymorphism clone of the generic poset with strict ordering has automatic homeomorphicity with respect to the class of polymorphism clones of countable \(\omega \) -categorical structures. Our results extend and generalize previous results by Bodirsky, Pinsker, and Pongrácz. PubDate: 2018-04-20 DOI: 10.1007/s00012-018-0504-1 Issue No:Vol. 79, No. 2 (2018)

Authors:Mike Behrisch; Edith Vargas-García Abstract: C-clones are polymorphism sets of so-called clausal relations, a special type of relations on a finite domain, which first appeared in connection with constraint satisfaction problems in work by Creignou et al. from 2008. We completely describe the relationship regarding set inclusion between maximal C-clones and maximal clones. As a main result we obtain that for every maximal C-clone there exists exactly one maximal clone in which it is contained. A precise description of this unique maximal clone, as well as a corresponding completeness criterion for C-clones is given. PubDate: 2018-04-20 DOI: 10.1007/s00012-018-0497-9 Issue No:Vol. 79, No. 2 (2018)