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)

Authors:Hajnal Andréka; Steven Givant Abstract: A measurable relation algebra is a relation algebra in which the identity element is a sum of atoms that can be measured in the sense that the “size” of each such atom can be defined in an intuitive and reasonable way (within the framework of the first-order theory of relation algebras). A large class of examples of such algebras, using systems of groups and coordinated systems of isomorphisms between quotients of the groups, has been constructed. This class of group relation algebras is not large enough to exhaust the class of all measurable relation algebras. In the present article, the class of examples of measurable relation algebras is considerably extended by adding one more ingredient to the mix: systems of cosets that are used to “shift” the operation of relative multiplication. It is shown that, under certain additional hypotheses on the system of cosets, each such coset relation algebra with a shifted operation of relative multiplication is an example of a measurable relation algebra. We also show that the class of coset relation algebras does contain examples that are not representable as set relation algebras. In later articles, it is shown that the class of coset relation algebras is adequate to the task of describing all measurable relation algebras in the sense that every atomic measurable relation algebra is essentially isomorphic to a coset relation algebra, and the class of group relation algebras is similarly adequate to the task of representing all measurable relation algebras in which the associated groups are finite and cyclic. PubDate: 2018-04-20 DOI: 10.1007/s00012-018-0516-x Issue No:Vol. 79, No. 2 (2018)

Authors:James Emil Avery; Jean-Yves Moyen; Pavel Růžička; Jakob Grue Simonsen Abstract: We consider the partition lattice \(\Pi (\lambda )\) on any set of transfinite cardinality \(\lambda \) and properties of \(\Pi (\lambda )\) whose analogues do not hold for finite cardinalities. Assuming AC, we prove: (I) the cardinality of any maximal well-ordered chain is always exactly \(\lambda \) ; (II) there are maximal chains in \(\Pi (\lambda )\) of cardinality \(> \lambda \) ; (III) a regular cardinal \(\lambda \) is strongly inaccessible if and only if every maximal chain in \(\Pi (\lambda )\) has size at least \(\lambda \) ; if \(\lambda \) is a singular cardinal and \(\mu ^{< \kappa } < \lambda \le \mu ^\kappa \) for some cardinals \(\kappa \) and (possibly finite) \(\mu \) , then there is a maximal chain of size \(< \lambda \) in \(\Pi (\lambda )\) ; (IV) every non-trivial maximal antichain in \(\Pi (\lambda )\) has cardinality between \(\lambda \) and \(2^{\lambda }\) , and these bounds are realised. Moreover, there are maximal antichains of cardinality \(\max (\lambda , 2^{\kappa })\) for any \(\kappa \le \lambda \) ; (V) all cardinals of the form \(\lambda ^\kappa \) with \(0 \le \kappa \le \lambda \) occur as the cardinalities of sets of complements to some partition \(\mathcal {P} \in \Pi (\lambda )\) , and only these cardinalities appear. Moreover, we give a direct formula for the number of complements to a given partition. Under the GCH, the cardinalities of maximal chains, maximal antichains, and numbers of complements are fully determined, and we provide a complete characterisation. PubDate: 2018-04-20 DOI: 10.1007/s00012-018-0514-z Issue No:Vol. 79, No. 2 (2018)

Authors:Daniele Mundici Abstract: A classical theorem of Jónsson and Tarski provides a sufficient condition for a generating set of an algebra to be free generating. The special case of the theorem for MV-algebras can also be proved with the help of techniques that work for algebras outside the scope of the Jónsson–Tarski theorem, and yield the recognition of free generating sets of free n-generator lattice ordered abelian groups. PubDate: 2018-04-18 DOI: 10.1007/s00012-018-0511-2 Issue No:Vol. 79, No. 2 (2018)

Authors:Ivan Chajda; Martin Goldstern; Helmut Länger Abstract: Let \({\mathcal {K}}\) be a congruence distributive variety and call an algebra hereditarily directly irreducible (HDI) if every of its subalgebras is directly irreducible. It is shown that every homomorphism from a finite direct product of arbitrary algebras from \({\mathcal {K}}\) to an HDI algebra from \({\mathcal {K}}\) is essentially unary. Hence, every homomorphism from a finite direct product of algebras \({\mathbf {A}}_i\) ( \(i\in I\) ) from \({\mathcal {K}}\) to an arbitrary direct product of HDI algebras \({\mathbf {C}}_j\) ( \(j\in J\) ) from \({\mathcal {K}}\) can be expressed as a product of homomorphisms from \({\mathbf {A}}_{\sigma (j)}\) to \({\mathbf {C}}_j\) for a certain mapping \(\sigma \) from J to I. A homomorphism from an infinite direct product of elements of \({\mathcal {K}}\) to an HDI algebra will in general not be essentially unary, but will always factor through a suitable ultraproduct. PubDate: 2018-04-18 DOI: 10.1007/s00012-018-0517-9 Issue No:Vol. 79, No. 2 (2018)

Authors:Paolo Lipparini Abstract: Suppose throughout that \(\mathcal V\) is a congruence distributive variety. If \(m \ge 1\) , let \( J _{ \mathcal V} (m) \) be the smallest natural number k such that the congruence identity \(\alpha ( \beta \circ \gamma \circ \beta \dots ) \subseteq \alpha \beta \circ \alpha \gamma \circ \alpha \beta \circ \dots \) holds in \(\mathcal V\) , with m occurrences of \( \circ \) on the left and k occurrences of \(\circ \) on the right. We show that if \( J _{ \mathcal V} (m) =k\) , then \( J _{ \mathcal V} (m \ell ) \le k \ell \) , for every natural number \(\ell \) . If \( J _{ \mathcal V} (1)=2 \) , that is, \(\mathcal V\) is 3-distributive, then \( J _{ \mathcal V} (m) \le m \) , for every \(m \ge 3\) . If \(\mathcal V\) is m-modular, that is, congruence modularity of \(\mathcal V\) is witnessed by \(m+1\) Day terms, then \( J _{ \mathcal V} (2) \le J _{ \mathcal V} (1) + 2m^2-2m -1 \) . Open problems are stated at various places. PubDate: 2018-04-18 DOI: 10.1007/s00012-018-0491-2 Issue No:Vol. 79, No. 2 (2018)

Authors:George M. Bergman Abstract: Marek Kuczma asked in 1980 whether for every positive integer n, there exists a subsemigroup M of a group G, such that G is equal to the n-fold product \(M\,M^{-1} M\,M^{-1} \ldots \,M^{(-1)^{n-1}}\) , but not to any proper initial subproduct of this product. We answer his question affirmatively, and prove a more general result on representing a certain sort of relation algebra by a family of subsets of a group. We also sketch several variants of the latter result. PubDate: 2018-04-17 DOI: 10.1007/s00012-018-0488-x Issue No:Vol. 79, No. 2 (2018)

Authors:G. Grätzer; H. Lakser Abstract: In the second edition of the congruence lattice book, Problem 22.1 asks for a characterization of subsets Q of a finite distributive lattice D such that there is a finite lattice L whose congruence lattice is isomorphic to D and under this isomorphism Q corresponds the the principal congruences of L. In this note, we prove some preliminary results. PubDate: 2018-04-17 DOI: 10.1007/s00012-018-0487-y Issue No:Vol. 79, No. 2 (2018)

Authors:Hans-E. Porst Abstract: Commutative varieties provide a natural setting for generalizing Hopf algebra theory over commutative rings, since they satisfy the various conditions identified in the category theoretical analysis of this theory to guarantee for example the existence of all naturally occurring forgetful functors in this context, the existence of universal measuring comonoids and the existence of generalized finite duals. It will be shown in addition, that crucial properties of the latter, known from the case of Hopf algebra theory over commutative rings, can be generalized Hopf algebra theory over a commutative variety. The attempt to generalize its construction leads to a couple of questions concerning properties of the monoidal structure of a commutative variety, which seem to be of a more general interest. PubDate: 2018-04-17 DOI: 10.1007/s00012-018-0500-5 Issue No:Vol. 79, No. 2 (2018)