Generation of positive lower-potent half-congruences (Q1281301)

Let \(S\) be a semigroup. A relation \(\xi\) on \(S\) is called positive if \(a\xi ab\) and \(a\xi ba\) for all \(a,b\in S\). \(\xi\) is compatible if \(a\xi b\) implies \(ac\xi bc\) and \(ca\xi cb\) for all \(c\in S\). The main theorem of this paper is concerned with reflexive positive relations. Let \(\pi\) denote a reflexive positive relation on \(S\), and \(\overline\pi\) the transitive closure of \(\pi\). The theorem says: \(\overline\pi\) is compatible if and only if for all \(a,b,c,d\in S\), \(a\pi c\) and \(b\pi d\) implies that there is \(u\in S\) satisfying the conditions \(a\mid u\) and \(b\mid u\) and \(u\overline\pi cd\). In particular in case \(\pi\) is a positive quasi-order on \(S\), then \(\overline\pi\) is compatible if and only if for all \(a,b,c\in S\) \(a\pi c\) and \(b\pi c\Rightarrow(\exists u\in S)\) \(a\mid u\) and \(b\mid u\) and \(u\overline\pi c\). By the results of this paper, the proofs of the reviewer's results [in Semigroup Forum 4, 255-261 (1972; Zbl 0261.20058) and in Proc. Am. Math. Soc. 41, 75-79 (1973; Zbl 0275.20106)] become very simple and improved.