Free strict inverse semigroups
An inverse semigroup is ``strict'' if it is a subdirect product of Brandt semigroups and groups. Strict inverse semigroups form a variety of inverse semigroups. Other than the varieties of Clifford semigroups, its subvarieties are those of the form \({\mathbf H}\vee{\mathbf B}\), where \(\mathbf H\) is some variety of groups and \(\mathbf B\) is the variety of combinatorial strict inverse semigroups, generated by the 5-element combinatorial Brandt semigroup. [See \textit{M. Petrich}, Inverse Semigroups (Wiley, 1984; Zbl 0546.20053).] The free strict inverse semigroups were first described by \textit{N. R. Reilly} [J. Lond. Math. Soc., II. Ser. 39, 102-120 (1989; Zbl 0636.20032)] and then by \textit{S. W. Margolis}, \textit{J. C. Meakin} and \textit{J. B. Stephen} [Can. J. Math. 42, 1084-1097 (1990; Zbl 0726.20040)]. The author offers an alternative structure theorem that is based on his work on the structure of semigroups whose congruence lattices satisfy certain conditions, using a construction of strict inverse semigroups in terms of the poset of \(\mathcal J\)-classes, the principal factors and certain partial homomorphisms between them (essentially equivalent to that of \textit{B. M. Schein} [see Petrich, above]). The free combinatorial strict inverse semigroup is explicitly described in these terms, those for the varieties \({\mathbf H}\vee{\mathbf B}\) less explicitly. In a sequel [J. Lond. Math. Soc., II. Ser. 45, 491-507 (1992; Zbl 0759.20016)] the author has refined his construction to provide an explicit one of the free strict inverse semigroup itself. The primary advantage of his method over the others is that his general structure theorems apply to wide classes of regular semigroups as well as inverse semigroups and can therefore be applied to the structure of various ``\(e\)-free'' and ``*-free'' regular semigroups.
variety of combinatorial strict inverse semigroups
5-element combinatorial Brandt semigroup
scientific article; zbMATH DE number 202981