On the structure of certain po-semigroups (Q1185774)

This article continues the work of \textit{A. M. Fourozanfar} and \textit{J. S. Pym} on the existence and structure of the minimal ideal in semigroups defined on partially ordered sets [ibid. 42, No. 2, 175-188 (1991; Zbl 0739.22003)]. The author calls a non-empty partially ordered set a complete semilattice if every non-empty subset has a supremum and every non-empty chain has an infimum. A semigroup on a partially ordered set is called a po-semigroup if its multiplication is non-decreasing (i.e., \(a\leq b\) and \(c\leq d\) always implies \(ac\leq bd\)). In analogy to the First Fundamental Theorem for compact semigroups with one-sidedly continuous multiplication the author shows that every po-semigroup \(S\) on a complete semilattice has minimal idempotents; from this fact he deduces in the usual way the existence and structure of the minimal ideal of \(S\).