| Summary: | By an associate inverse subsemigroup of a regular semigroup $S$ we mean a subsemigroup $T$ of $S$ containing a least associate of each $x \in S$, in relation to the natural partial order $\leq_S$ in $S$. In this paper we investigate a class of orthodox monoids with an associate inverse subsemigroup and obtain a known description of uniquely unit regular orthodox semigroups as a corollary. Also, by considering a more general situation, we identify the homomorphic image of a kind of semidirect product of a band with identity by an inverse monoid, thus extending a known result for unit regular orthodox semigroups.
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