In this paper we describe, in a purely algebraic language, truthcomplete finite-valued propositional logical calculi extending the classical Boolean calculus. We also give a new proof of the Completeness Theorem for such calculi. We investigate the quasi-varieties of algebras playing an analogous role in the theory of these finite-valued logics to the role played by the variety of Boolean algebras in classical logic.

We analyze the two-kaon Bose-Einstein correlation in the reactions of 160A GeV/c P b t P b within an event generator URASiMA. The apparent radius of kaon-source is extracted. The theoretical result agrees quite well with the preliminary data of NA44.

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Fuchs [i] posed the following problem:investigate those Abelian groups in which each infinite subgroup B is embedded in a direct summand of cardinality IBI (Problem No. 5).Naturally, the questions related to this problem are of interest not only for groups, but for modules over various rings. It turns out that the structure of the groups and modules with the above property can essentially depend not only on the cardinality of the considered object, but on the choice of an axiomatic basis for set theory.[If the axiomatic basis of set theory is not specified, we tacitly take it to be ZFC (Zermelo-Fraenkel with the Axiom of Choice), to which we add, if necessary, various set-theoretic hypotheses.]For example, in studying the class of k-separable torsion-free Abelian groups (for the definition see, e.g., [2, 3]), which is close to the class of free Abelian groups, it was found that: i) for singular and for regular, weakly compact cardinals k, any k-separable torsion-free Abelian group of cardinality k is free (see, e.g., It is our goal in this present paper to show that under the assumption V = L any ring R in a very extensive class of rings can be realized (modulo so-called k-small endomorphisms) as a ring of endomorphisms of each of a family of 2 k pairwise nonisomorphic, torsionfree, Abelian groups of cardinality k that are also R-modules with "very strong local decomposability into direct sums" (Theorems 1 and 2). As corollaries (for any of the above-mentioned uncountable regular cardinals k), we will establish (under the assumption V = L) the existence of families of 2 k pairwise nonisomorphic R-modules of cardinality k in which each subset~ of cardinality less than k can be embedded in some direct summand~rof cardinality less than k, but no R-module in these families is decomposable into a direct sum of even two R-submodules of cardinality k (Corollaries i and 2). Note that some results for modules of singular cardinality k, contrasting with those obtained in this present paper for regular cardinals k, were found by Hodges [7, p. 218].In the present paper, unless stated otherwise, "R-module" will mean "unital left Rmodule," and "group" will mean "Abelian group." *In memory of A. I. Kokorin