Associativity of the regular semidirect product of existence varieties

Abstract

The associativity of the regular semidirect product of existence varieties introduced by Jones and Trotter was proved under certain conditions by Reilly and Zhang. Here we establish associativity in many new cases. Moreover, we prove that the regular semidirect product is right distributive with respect to the join operation. In particular, both associativity and right distributivity yield within the varieties of completely simple semigroups. Analogous results are obtained for e-pseudovarieties of finite regular semigroups.2000 Mathematics subject classification: primary 20M17,20M07.

Rees matrix semigroups and the regular semidirect product

“…By applying the main result in [4], we can deduce the following result which handles the case not covered by Theorem 3.9. Given a variety U of inverse semigroups and a variety V of completely simple semigroups with WB c V, the main result Theorem 3.9 of the previous section allows us to construct a model of any bifree object in the e-variety U * r V by making use of the Pastijn-Rees product construction.…”

“…By applying the main result in [4], we can deduce the following result which handles the case not covered by Theorem 3.9.…”

“…The following result from [4] describes the regular elements of semidirect products of regular semigroups, and gives inverses. [5] Rees matrix semigroups 43…”

Comparing the regular and the restricted regular semidirect products

“…Throughout, for a class C of regular semigroups, C denotes the smallest e-variety containing C. We call U * r V the regular semidirect product of the e-varieties U and V. For various results on * r , the reader is referred to Jones and Trotter [12]. Concerning associativity of the partial product * r , the most advanced result is due to Billhardt and the second author [8]. In particular, * r is associative inside LI.…”