Algebraic properties of Zappa-Szép products of semigroups and monoids
Suppose \(S\) and \(T\) are semigroups, with \(T\) acting on \(S\) from the left by \((t,s) \mapsto t \cdot s\) and \(S\) acting on \(T\) from the right by \((t,s) \mapsto t^s\), and the actions connected by the rules \(t \cdot (ss') = (t\cdot s)(t^s \cdot s')\) and \((tt')^s = t^{t' \cdot s} t^{'s}\). Under the product \((s,t)(s', t') = (s (t \cdot s'), t^{s'}t')\), \(S \times T\) becomes the Zappa-Szép product \(S \bowtie T\). If \(S\) and \(T\) are monoids, then under natural further assumptions, the product is again a monoid. This semigroup product was studied in depth by \textit{M. Kunze} [Acta Math. Hung. 41, 225--239 (1983; Zbl 0538.20033)] and special cases have been considered since then. The author shows that, given a left restriction semigroup \(S\) with semilattice \(E\) of projections, there are natural actions that yield the Zappa-Szép product \(E \bowtie S\), having \(S\) as a retract in a natural way and containing the semilattice \(\bar{E} = \{(e,e)\}\) isomorphic to \(E\). Although \(E \bowtie S\) is not itself left restriction, it contains a maximum left restriction subsemigroup having semilattice of projections \(\bar{E}\), which again retracts on \(S\).
scientific article; zbMATH DE number 6893037
scientific article