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Variable Varieties of Semigroups
by
Tatjana Petkovic
University of Nis, Nis, Yugoslavia and Turku Centre for Computer Science, Turku, Finland
Coauthors: Miroslav \'Ciric (University of Nis), Stojan Bogdanovic (University of Nis)
An identity over an alphabet A is a pair of words from the free semigroup A+ which is usually written as a formal equality of these words. A semigroup S is said to satisfy a set of identities \Sigma over A if the kernel of each homomorphism from A+ into S contains \Sigma. But if the kernel of each homomorphism from A+ into S contains a non-trivial identity from \Sigma, then we say that S satisfies variabily \Sigma, or that it satisfies \Sigma as a variable identity , and a class of semigroups determined by a variable identity \Sigma is called a variable variety and denoted by [\Sigma]v. This is the same concept which was introduced by Putcha and Weissglass (1971, 1972), but the definition given here is closer to the definition of ordinary identities than the one of Putcha and Weissglass. In a way, this concept traces one's origin to the concept of pseudo identities and pseudo varieties, introduced by Schein in the 1960's (or disjunctive identities and varieties, as they were called by Mashevitzky). The related concepts, the so-called inclusive identities and collective identities, have been studied by Lyapin (1975, 1982), Almeida (1989) and Mashevitzky (1996).
Putcha and Weissglass characterized the classes of periodic semigroups which are nilpotent extensions of unions of groups and semilattices of groups as variable varieties. Here we use variable identities to describe the classes of periodic semigroups which are nil-extensions and retractive nil-extensions of unions of groups in the general and various special cases. The obtained results generalize those of Putcha and Weissglass, as well as the results of Bell (1977) concerning rings satisfying variable semigroup identities, and the results of \'Ciri\'c and Bogdanovi\'c (1994) concerning semigroups satisfying some ordinary identities.
On the other hand, we show that variable identities can be also very useful in study of semigroups having some properties as hereditary ones. We consider the class Her(RIS) of all semigroups whose every subsemigroup belongs to the class RIS of semigroups in which the radical of every ideal is a subsemigroup, we characterize Her(RIS) as a variable variety and prove that it consists of all semigroups which are not divided by the five element Brandt semigroup B2. We also show that there are 6 maximal solutions of the inequality [\Sigma]v subset or equal Her(RIS) and that one of them is the greatest solution of the equation [\Sigma]v = Her(RIS).
Date received: May 6, 2000
Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caec-19.