Decompositions of semigroups.

In this article we discuss the factorisations of semigroups and monoids in the context of direct, semidirect and Zappa–Szép products addressing the question of uniqueness. An equivalence between external and internal Zappa–Szép product of groups and monoids is known, but no such correspondence exists for semigroups in general. We prove the equivalence between external and internal Zappa–Szép product of semigroups subject to certain conditions in this article. We end with some illustrative examples of the Zappa–Szép product of the bisimple inverse monoids. [ABSTRACT FROM AUTHOR]