1,721,031 research outputs found
Relation identities in 3-distributive varieties
Full text linkLet alpha, beta, gamma, ... Theta, Psi, ... R, S, T, ... be variables for, respectively, congruences, tolerances and reflexive admissible relations. Let juxtaposition denote intersection. We show that if the identityalpha(beta omicron Theta) subset of alpha beta omicron alpha Theta omicron alpha betaholds in a variety V, then V has a majority term, equivalently, V satisfies alpha(beta omicron gamma ) subset of alpha beta omicron alpha gamma. The result is unexpected, since in the displayed identity we have one more factor on the right and, moreover, if we let Theta be a congruence, we get a condition equivalent to 3-distributivity, which is well-known to be strictly weaker than the existence of a majority term. The above result is optimal in many senses; for example, we show that slight variations on the displayed identity, such as R(S omicron gamma) subset of RS omicron R gamma omicron RS or R(S omicron T) subset of RS omicron RT omicron RT omicron RS hold in every 3-distributive variety, hence do not imply the existence of a majority term. Similar identities are valid even in varieties with 2 Gumm terms, with no distributivity assumption. We also discuss relation identities in n-permutable varieties and present a remark about implication algebras
Some transfinite natural sums
Full text linkWe 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. (c) 2018 WILEY-VCH Verlag GmbH & Co. KGaA, Weinhei
The Gumm level equals the alvin level in congruence distributive varieties
Full text linkCongruence modular and congruence distributive varieties can be characterized by the existence of sequences of Gumm and Jónsson terms, respectively. Such sequences have variable lengths, in general. It is immediate from the above paragraph that there is a variety with Gumm terms but without Jónsson terms. We prove the unexpected result that, on the other hand, if some variety has both kinds of terms, then the minimal lengths of the sequences differ at most by 1. It follows that every congruence distributive variety with (r+1) Day terms has (r^2-r+3) Jónsson terms
A variety V is congruence modular if and only if V satisfies U(R o R) c (UR)^h, for some h
Full text linkWe present a characterization of congruence modularity by means of an identity involving a tolerance and a reflexive and admissible relation
A very general covering property
Full text linksummary:We introduce a general notion of covering property, of which many classical definitions are particular instances. Notions of closure under various sorts of convergence, or, more generally, under taking kinds of accumulation points, are shown to be equivalent to a covering property in the sense considered here (Corollary 3.10). Conversely, every covering property is equivalent to the existence of appropriate kinds of accumulation points for arbitrary sequences on some fixed index set (Corollary 3.5). We discuss corresponding notions related to sequential compactness, and to pseudocompactness, or, more generally, properties connected with the existence of limit points of sequences of subsets. In spite of the great generality of our treatment, many results here appear to be new even in very special cases, such as -compactness and -pseudocompactness, for an ultrafilter, and weak (quasi) -(pseudo)-compactness, for a set of ultrafilters, as well as for -compactness, with and ordinals
The distributivity spectrum of Baker’s variety
Full text linkFor every n, we evaluate the smallest k such that the congruence inclusion α(β∘nγ)⊆αβ∘kαγ 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
The Tschantz and the alvin higher conditions are equivalent in congruence distributive varieties
Full text linkWe show that, under the assumption of congruence distributivity, a condition by S. Tschantz characterizing congruence modularity is equivalent to a variant of the classical Jonsson condition. Here equivalence is intended in a strong sense, to the effect that the corresponding sequences of terms have exactly the same lengt
Another characterization of congruence distributive varieties
Full text linkWe provide a Maltsev characterization of congruence distributive varieties by showing that a variety is congruence distributive if and only if the congruence identity α∩(β∘γ∘β)⊆αβ∘γ∘αβ∘γ ... (k factors) holds in , for some natural number k
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