Semigroups of inverse quotients. (Q2392042)

The paper discusses the notion of left I-quotients in inverse semigroups. A subsemigroup \(S\) of an inverse semigroup \(Q\) is called a left I-order in \(Q\) (and \(Q\) is a semigroup of left I-quotients of \(S\)) if every \(q\in Q\) can be written as \(q=a^{-1}b\) where \(a,b\in S\). In Section~2 the authors extend the result of \textit{J.~Fountain} and \textit{M.~Petrich} [Math. Proc. Camb. Philos. Soc. 98, 413-426 (1985; Zbl 0589.20042)] by showing that Brandt semigroups of left I-quotients of a given semigroup are unique up to isomorphism. Section~3 focuses on left ample semigroups, i.e.\ on semigroups \(S\) which embed into an inverse semigroup in such a way that if \(a\in S\) then \(aa^{-1}\in S\). Theorem~3.7 states that a left ample semigroup \(S\) is an I-order in its inverse hull if and only if it satisfies \[ \text{for any }a,b\in S,\;Sa\cap Sb=Sc\text{ for some }c \in S.\tag{LC} \] In Section~4 the authors investigate semigroups which are strong semilattices of left ample semigroups having (LC), and they extend the results of \textit{R.~L.~Gantos} [Q. J. Math., Oxf. II. Ser. 22, 379-393 (1971; Zbl 0247.20070)]. Finally, in Section~5 they prove that the category of left ample semigroups with (LC) is equivalent to the category of bisimple inverse semigroups.