The structure of \(\mathcal L^*\)-inverse semigroups.
A semigroup is `abundant' if each class of the `potential' Green's relations \({\mathcal L}^*\) and \({\mathcal R}^*\) contains an idempotent. As common generalizations of regular semigroups and cancellative monoids, they have attracted considerable effort devoted to generalizing theorems on particular classes of regular semigroups to appropriate classes of abundant semigroups. For instance, the class of `type A' (now frequently termed `ample') semigroups extends the class of inverse semigroups appropriately, although a somewhat technical condition known as idempotent-connectedness is frequently needed for explicit generalizations of particular theorems. The aim of this paper is to generalize a theorem of \textit{M. Yamada} [Semigroup Forum 6, 113-128 (1973; Zbl 0255.20038)] on left inverse semigroups (regular semigroups in which the idempotents form a left regular band). The \({\mathcal L}^*\)-inverse semigroups are defined to be the abundant semigroups that are idempotent-connected and in which the idempotents once again form a left regular band. The concept of a left wreath product of a left regular band and a type A semigroup is defined and it is proven that the \({\mathcal L}^*\)-inverse semigroups correspond precisely to the class of all such wreath products.
scientific article; zbMATH DE number 5118315
scientific article