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- W2474967395 abstract "Semigroups of varieties have first been studied in the case of groups, where Neumann, Neumann and Neumann-and independently Smel'kin-have discovered a surprising result: The semigroup of varieties of groups is a free monoid with zero ([13], [14], [18]). Since then various authors have investigated semigroups of varieties of group-like structures such as quasigroups, rings, lattice-ordered groups and Lie-algebras ([2], [5], [8], [15], [19]). It was Mal'cev who considered the general case, and he succeeded in giving a sufficient condition under which the subvarieties of a given variety form a semigroup [7]. Exploiting this idea Kbhler studied the semigroup of varieties of Brouwerian algebras and Blok and Kohler did this for the semigroup of varieties of generalized interior algebras ([6], [1]). Nearly all the cited papers centered around the question whether a result similar to the one for groups could be obtained. This paper continues these efforts in giving a positive answer to the question above for the variety of Brouwerian semilattices. The paper is divided into two parts. The first one introduces the notion of an extension of a Brouwerian semilattice. Based on ideas originally introduced by Nemitz [10] and most elegantly generalized by Sclhmidt ([16], [17]) it is proven that every extension of a Brouwerian semilattice by another can be imbedded into some special kind of extension which we call strongly splitting extension. The second part introduces the multiplication of varieties of Brouwerian semilattices, thus giving the set of varieties a semigroup structure. Based on results from ? 1 and using techniques originally developed in [6] it is finally proven that the semigroup of varieties is a free monoid with zero." @default.
- W2474967395 created "2016-07-22" @default.
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- W2474967395 date "2016-01-01" @default.
- W2474967395 modified "2023-09-24" @default.
- W2474967395 title "THE SEMIIGROUP OF VARIETIES OF BROUWERIAN SEMILATHCES" @default.
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