On P-Q ordered semigroups
A partially ordered semigroup \(S\) is called P-ordered if for any \(a,b \in S\), \(ab \geq b\). The semigroup \(S\) is called Q-ordered if \(a \leq b\) implies \(a = b\) or \(b = ac\) for some \(c \in S\). These are the duals of N- M-ordered semigroups [in \textit{S. Y. Kwan} and \textit{K. P. Shum}, Semigroup Forum 19, 151-175 (1980; Zbl 0438.06010)]. The authors prove a number of results on P-Q-ordered semigroups. Theorem 1: Let \(S\) be a P-ordered semigroup. If \(S\) is regular, then \(S\) is a band. If \(S\) is an inverse semigroup, then \(S\) is a semilattice. Theorem 2: Let \(S\) be a P-Q-ordered semigroup in which every right principal ideal is completely prime, then \(S\) has no elements of finite order.
scientific article; zbMATH DE number 698623