Words on free bands with inverse transversals.
An inverse transversal \(T\) of a regular semigroup \(S\) is an inverse subsemigroup with the property that each element \(a\) of \(S\) has a unique inverse \(a^0\) belonging to \(T\). The unary operation so defined turns \(S\) into a regular unary semigroup. The first author [Commun. Algebra 36, No. 7, 2487-2502 (2008; Zbl 1157.20036)] found a set of identities that define the regular semigroups with inverse transversal within the variety of all regular unary semigroups. Their algebraic study had originated with \textit{T. S. Blyth} and \textit{R. McFadden} [Proc. R. Soc. Edinb., Sect. A, Math. 92, 253-270 (1982; Zbl 0507.20026)]. The purpose of this paper is to solve the word problem for the free objects in the subvariety consisting of the bands with inverse (that is, semilattice) transversal. While the solution might have been achieved more simply by adapting the standard methods for the free band itself, they are sufficiently practical to enable calculation of the cardinalities of the \(\mathcal D\)-classes, for instance. Integral to their solution is the solution to the word problem in the subvariety of right regular bands with semilattice transversal.
scientific article; zbMATH DE number 6518141
scientific article