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A variety is a class of algebraic structures axiomatized by a set of equations. An equation is basic if there is at most one occurrence of a nonnullary operation symbol on each side. We show that a variety axiomatized by basic equations has the strong amalgamation property. Suppose further that the language has no constant symbol and, for each equation, either one side is operation-free, or exactly the same variables appear on both sides. Then also the joint embedding property holds, hence if the language is finite we get the existence of Fraïssé limits for finite algebras. Examples include virtually all the varieties defining classical Maltsev conditions. In a few special cases, the above properties are preserved when further unary operation symbols appear in the equations. [ABSTRACT FROM AUTHOR]

We solve some problems about relative lengths of Maltsev conditions, in particular, we provide an affirmative answer to a classical problem raised by A. Day more than 50 years ago. In detail, both congruence distributive and congruence modular varieties admit Maltsev characterizations by means of the existence of a finite but variable number of appropriate terms. A. Day showed that from Jónsson terms t 0 , ... , t n witnessing congruence distributivity it is possible to construct terms u 0 , ... , u 2 n - 1 witnessing congruence modularity. We show that Day's result about the number of such terms is sharp when n is even. We also deal with other kinds of terms, such as alvin, Gumm, directed, as well as with possible variations we will call "specular" and "defective". All the results hold when restricted to locally finite varieties. [ABSTRACT FROM AUTHOR]

We say that an idempotent term t is an exact-m-majority term if t evaluates to a, whenever the element a occurs exactly m times in the arguments of t, and all the other arguments are equal. If m < n and some variety 퓥 has an n-ary exact-m-majority term, then 퓥 is congruence modular. For certain values of n and m, for example, n = 5 and m = 3, the existence of an n-ary exact-m-majority term neither implies congruence distributivity, nor congruence permutability. [ABSTRACT FROM AUTHOR]

We show that for a large class of varieties of algebras, the equational theory of the congruence lattices of the members is not finitely based. [ABSTRACT FROM AUTHOR]

We devise a condition strictly between the existence of an n-ary and an n + 1-ary near-unanimity term. We evaluate exactly the distributivity and modularity levels implied by such a condition. [ABSTRACT FROM AUTHOR]

Mitschke showed that a variety with an m-ary near-unanimity term has Jónsson terms t 0 , ⋯ , t 2 m - 4 witnessing congruence distributivity. We show that Mitschke's result is sharp. We also evaluate the best possible number of Day terms witnessing congruence modularity. More generally, we characterize exactly the best bounds for many congruence identities satisfied by varieties with an m-ary near-unanimity term. Finally we present some simple observations about terms with just one "dissenter", a generalization of a minority term. [ABSTRACT FROM AUTHOR]

Contrary to the finitary case, the set Г A) of all the non-generators of an infinitary algebra A is not necessarily a subalgebra of A. We show that the phenomenon is ubiquitous: every algebra with at least one infinitary operation can be embedded into some algebra B such that Г (B) is not a subalgebra of B. As far as expansions are concerned, there are examples of infinite algebras A such that in every expansion B of A the set Г (B) is a subalgebra of B. However, under relatively weak assumptions on A, it is possible to get some expansion B of A such that Г(B) fails to be a subalgebra of B. [ABSTRACT FROM AUTHOR]

For every n, we evaluate the smallest k such that the congruence inclusion $\alpha (\beta \circ _n \gamma) \subseteq \alpha \beta \circ _{k} \alpha \gamma $ holds in a variety of reducts of lattices introduced by K. Baker. We also study varieties with a near-unanimity term and discuss identities dealing with reflexive and admissible relations. [ABSTRACT FROM AUTHOR]

We show that, under the assumption of congruence distributivity, a condition by S. Tschantz characterizing congruence modularity is equivalent to a variant of the classical Jónsson condition. Here equivalence is intended in a strong sense, to the effect that the corresponding sequences of terms have exactly the same length. [ABSTRACT FROM AUTHOR]

We study a transfinite iteration of the ordinal Hessenberg natural sum obtained by taking suprema at limit stages. We show that such an iterated natural sum differs from the more usual transfinite ordinal sum only for a finite number of iteration steps. The iterated natural sum of a sequence of ordinals can be obtained as a mixed sum (in an order‐theoretical sense) of the ordinals in the sequence; in fact, it is the largest mixed sum which satisfies a finiteness condition. We introduce other infinite natural sums which are invariant under permutations and show that all the sums under consideration coincide in the countable case. [ABSTRACT FROM AUTHOR]

As far as algebraic properties are concerned, the usual addition on the class of ordinal numbers is not really well behaved; for example, it is not commutative, nor left cancellative etc. In a few cases, the natural Hessenberg sum is a better alternative, since it shares most of the usual properties of the addition on the naturals. A countably infinite iteration of the natural sum has been used in a recent paper by Väänänen and Wang, with applications to infinitary logics. We present a detailed study of this infinitary operation, showing that there are many similarities with the ordinary infinitary sum, providing an order theoretical characterization, and finding connections with certain kinds of infinite mixed sums. [ABSTRACT FROM AUTHOR]

We show that two possible definitions of sequential pseudocompactness are equivalent, and point out some consequences. [ABSTRACT FROM AUTHOR]

Abstract: We characterize various Menger/Rothberger-related properties, and discuss their behavior with respect to products. [Copyright &y& Elsevier]

Abstract: We introduce a covering notion depending on two cardinals, which we call -compactness, and which encompasses both pseudocompactness and many other known generalizations of pseudocompactness. For Tychonoff spaces, pseudocompactness turns out to be equivalent to -compactness. We provide several characterizations of -compactness, and we discuss its connection with D-pseudocompactness, for D an ultrafilter. The connection turns out to be rather strict when the above notions are considered with respect to products. In passing, we provide some conditions equivalent to D-pseudocompactness. Finally, we show that our methods provide a unified treatment both for -compactness and for -compactness. [Copyright &y& Elsevier]

We prove, in ZFC alone, some new results on regularity and decomposability of ultrafilters; among them: (a) If m ≥ 1 and the ultrafilter D is (m(λ+n), m(λ+n))-regular, then D is κ -decomposable for some κ with λ ≤ κ ≤ 2λ (Theorem 4.3(a')). (b) If λ is a strong limit cardinal and D is (m(λ+n), m(λ+n))-regular, then either D is (cf λ, cf λ)-regular or there are arbitrarily large κ < λ for which D is κ -decomposable (Theorem 4.3(b)). (c) Suppose that λ is singular, λ < κ, cf κ ≠ cf λ and D is (λ+, κ)-regular. Then: (i) D is either (cf λ, cf λ)-regular, or (λ', κ)-regular for some λ' < λ (Theorem 2.2). (ii) If κ is regular, then D is either (λ, κ)-regular, or (ω, κ')-regular for every κ' < κ (Corollary 6.4). (iii) If either (1) λ is a strong limit cardinal and λ<λ < 2κ, or (2) λ<λ < κ, then D is either λ -decomposable, or (λ', κ)-regular for some λ' < λ (Theorem 6.5). (d) If λ is singular, D is (μ, cf λ)-regular and there are arbitrarily large ν < λ for which D is ν -decomposable, then D is κ -decomposable for some κ with λ ≤ κ ≤ λ<μ (Theorem 5.1; actually, our result is stronger and involves a covering number). (e) D × D' is (λ, μ)-regular if and only if there is a ν such that D is (ν, μ)-regular and D' is (λ, ν')-regular for all ν∼ < ν (Proposition 7.1). We also list some problems, and furnish applications to topological spaces and to extended logics (Corollar-ies 4.6 and 4.8) (© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [ABSTRACT FROM AUTHOR]

The class of linearly ordered sets with one order preserving unary operation has the Strong Amalgamation Property (SAP). The class of linearly ordered sets with one strict order preserving unary operation has AP but not SAP. The class of linearly ordered sets with two order preserving unary operations does not have AP. For every set F , the class of linearly ordered sets with an F -indexed family of automorphisms has SAP. Corresponding results are proved in the case of order reversing operations. Various subclasses of the above classes are considered and some model-theoretical consequences are presented. [ABSTRACT FROM AUTHOR]

Abstract: We present instances of the following phenomenon: if a product of topological spaces satisfies some given compactness property then the factors satisfy a stronger compactness property, except possibly for a small number of factors. The first known result of this kind, a consequence of a theorem by A.H. Stone, asserts that if a product is regular and Lindelöf then all but at most countably many factors are compact. We generalize this result to various forms of final compactness, and extend it to two-cardinal compactness. In addition, our results need no separation axiom. [Copyright &y& Elsevier]

The article discusses the duality for compact logics and substitution in the abstract model theory. It studies the relationships between the abstract properties of a logic L and the accompanying algebraic properties of its relations. This relationship has been proven useful in the Abstract Model Theory which most often give the main tools for proving theorems. It defines the concept of substitution using an alternative definition of logics produced by quantifiers and applications.

An identity s = t is linear if each variable occurs at most once in each of the terms s and t. Let T be a tolerance relation of an algebra $${\mathcal{A}}$$ in a variety defined by a set of linear identities. We prove that there exist an algebra $${\mathcal{B}}$$ in the same variety and a congruence θ of $${\mathcal{B}}$$ such that a homomorphism from $${\mathcal{B}}$$ onto $${\mathcal{A}}$$ maps θ onto T. [ABSTRACT FROM AUTHOR]