Free orthodox semigroups and free bands with inverse transversals.
A regular semigroup \(S\) has an inverse transversal if it contains an inverse subsemigroup \(S^0\) with the property that each element \(a\) of \(S\) has a unique inverse, which may be denoted \(a^0\), in \(S^0\). The author denotes the class of such semigroups -- which he terms ist-semigroups -- by \(\mathcal{IST}\). They were introduced by \textit{T. S. Blyth} and \textit{R. McFadden} [Proc. R. Soc. Edinb., Sect. A 92, 253-270 (1982; Zbl 0507.20026)]. The operation \(a\to a^0\) turns such semigroups into regular unary semigroups. In [Commun. Algebra 36, No. 7, 2487-2502 (2008; Zbl 1157.20036)] the author showed that \(\mathcal{IST}\) is definable within the variety of regular unary semigroups by the identities \((a^0b^0)^0=b^{00}a^{00}\), \(aa^0bb^0aa^0=aa^0bb^0\) and \(a^0ab^0ba^0a=b^0ba^0a\). Somewhat analogous to the concept of e-variety (or bivariety) of regular semigroups is that of an ist-variety: a class of ist-semigroups closed under homomorphic images, ist-subsemigroups and direct products. For example, the orthodox ist-semigroups form an ist-variety. In this paper, the author introduces a concept of freeness, analogous to the e-free (or bifree) property in e-varieties, and then constructs the orthodox ist-semigroups that are free in this fashion. It is notable that their inverse transversals are free inverse semigroups. He also constructs the ist-bands that are free in this fashion. There remain interesting avenues of research to pursue along these lines.
scientific article; zbMATH DE number 5857721
scientific article