Algebraic properties of Zappa–Szép products of semigroups and monoids
Rida-e Zenab · Semigroup Forum · 2017
Direct, semidirect and Zappa–Szép products provide tools to decompose algebraic structures, with each being a natural generalisation of its predecessor. In this paper we examine Zappa–Szép products of monoids and semigroups and investigate generalised Greens relations $${\mathcal R}^{*},\, {\mathcal L}^{*},\, \widetilde{\mathcal {R}}_E$$ and $$\widetilde{\mathcal {L}}_E$$ for these Zappa–Szép products. We consider a left restriction semigroup S with semilattice of projections E and define left and right actions of S on E and E on S, respectively, to form the Zappa–Szép product $$E \bowtie S$$ . We further investigate properties of $$E \bowtie S$$ and show that S is a retract of $$E\bowtie S$$ . We also find a subset T of $$E \bowtie S$$ which is left restriction.