Associativity of products of existence varieties of regular semigroups
The concept of an e-variety -- a class of regular semigroups that is closed under products, quotients and regular subsemigroups -- has allowed some of the techniques of universal algebra to be applied to regular semigroups. Motivated by the theory of semigroup varieties, the reviewer and \textit{P. G. Trotter} [Trans. Am. Math. Soc. 349, No. 11, 4265-4310 (1997; Zbl 0892.20037)] introduced a partial operation on the set of all e-varieties, as follows: if \(\mathcal{U,V}\) are e-varieties, at least one of which is contained in the e-variety \(\mathcal{CS}\) of completely simple semigroups, then \({\mathcal U}*{\mathcal V}\) consists of the regular parts of the wreath products of members of \(\mathcal U\) with members of \(\mathcal V\). One direction taken by the paper under review is to extend the work of the cited paper and the authors' paper [Semigroup Forum 53, No. 1, 1-24 (1996; Zbl 0854.20069)] on the properties of this operation. Associativity of the product is proven under some broad conditions; for instance if \(\mathcal{U,V}\) are group varieties and \(\mathcal W\) is ``monoidal'', \({\mathcal U}*({\mathcal V}*{\mathcal W})=({\mathcal U}*{\mathcal V})*{\mathcal W}\). This enables properties of such products as \(\mathcal{CS}*{\mathcal V}\) to be found for various \(\mathcal V\). A second direction is to study a different ``wreath product'', introduced by the authors in the cited paper, related to certain Malcev products of e-varieties. Reviewer's remark: In a subsequent paper, \textit{B. Billhardt} and \textit{M. B. Szendrei} [Associativity of regular semidirect product of existence varieties (to appear)] have extended the main associativity result to a broader class of cases.
scientific article; zbMATH DE number 1308297
scientific article