Restriction semigroups and \(\lambda\)-Zappa-Szép products (Q682117)

Let \(S\) and \(T\) be semigroups and suppose we have maps \(T\times S\longrightarrow S, \quad (t,s)\mapsto t\cdot s\) and \(T\times S\longrightarrow T, \quad (t,s)\mapsto t^s\) such that for all \(s, s'\in S\) and \(t, t'\in T\) the following hold: (ZS1) \(tt'\cdot s=t(t'\cdot S)\), (ZS2) \(t\cdot (ss')=(t\cdot s)(t^s\cdot s')\), (ZS3) \((t^s)^{s'}={t}^{ss'}\), (ZS4) \((tt')^s=(t)^{t'\cdot s}{t'}^s\). The semigroup \(S\times T\) with a binary operation defined by \((s,t)(s',t')=(s(t\cdot s'),t^{s'}t')\) is called the \textit{(external) Zappa-Szép product} of \(S\) and \(T\) and denoted by \(S\bowtie T\). As the authors write in the introduction, ``restriction semigroups are a variety of binary semigroup, naturally extending the class of inverse semigroups. Our broad aim is to study restriction semigroups using semidirect products and Zappa-Szép products. [\dots] The semidirect product of two inverse semigroups is not inverse in general. The \(\lambda\)-semidirect product of two inverse semigroups is again inverse, see \textit{B. Bilhardt} [Semigroup Forum 45, 45--54 (1992; Zbl 0769.20027)]. It is clear that the Zappa-Szép product of two inverse semigroups is not inverse in general. [\dots] \textit{N. D. Gilbert} and \textit{S. Wazzan} [Semigroup Forum 77, 438--455 (2008; Zbl 1169.20031)] generalized Bilhardt's concept [loc. cit.] of \(\lambda\)-semidirect product to what they named as \(\lambda\)-Zappa-Szép product''. The present paper put the work of Bilhardt on \(\lambda\)-semidirect product [loc. cit.], and that of Gilbert and Wazzan \(\lambda\)-Zappa-Szép products [loc. cit.] into broader context of restriction semigroups.