Ordered semigroups whose elements are separated by prime ideals
intra-regular (regular) ordered semigroup
Let \(S\) be an ordered semigroup. \(S\) is called intra-regular if for every \(a\in S\) there exist \(x, y\in S\) such that \(a\leq xa^2y\). A subset \(T\) of \(S\) is called prime if the complement \(S\setminus T\) of \(T\) to \(S\) is either empty or a subsemigroup of \(S\). A nonempty subset \(T\) of \(S\) is called an ideal of \(S\) if (1) \(TS\subseteq T\), \(ST\subseteq T\) and (2) if for \(a\in T\), \(b\in S\) and \(b\leq a\), \(b\in T\). The ideal of \(S\) generated by \(a\in S\) is denoted by \(I(a)\). We say that the elements of \(S\) are separated by prime ideals of \(S\) if for all \(a,b\in S\) such that \(b\notin I(a)\) there exists a prime ideal \(P\) of \(S\) such that \(a\in P\) and \(b\notin P\). The main result of the paper is as follows: The elements of an ordered semigroup \(S\) are separated by prime ideals of \(S\) if and only if \(S\) is intra-regular. This result generalizes the corresponding result for semigroups (without order).
scientific article; zbMATH DE number 6034817
scientific article