Ordered inverse semigroups

Abstract:

In this paper, we consider two questions: one is to characterize the structure of ordered inverse semigroups and the other is to give a condition in order that an inverse semigroup is orderable. The solution of the first question is carried out in terms of three types of mappings. Two of these consist of mappings of an $\mathcal {R}$-class onto an $\mathcal {R}$-class, while one of these consists of mappings of a principal ideal of the semilattice $E$ constituted by idempotents onto a principal ideal of $E$. As for the second question, we give a theorem which extends a well-known result about groups that a group $G$ with the identity $e$ is orderable if and only if there exists a subsemigroup $P$ of $G$ such that $P \cup {P^{ - 1}} = G,P \cap {P^{ - 1}} = \{ e\}$ and $xP{x^{ - 1}} \subseteqq P$ for every $x \in G$.