Generalized Archimedean ideals. (English) Zbl 0579.22005

Let S be a semigroup and E denote the set of all idempotents in S. S is said to be semi-normal if \(eS=Se\) for all \(e\in E\). An ideal A in a compact semigroup S is called generalized Archimedean if x,y\(\in A\) implies \(\Gamma\) (x)\(\cap AyA\neq \emptyset\), where \(\Gamma\) (x) is the closure of the monothetic semigroup generated by x. The following two results are proved:
I. An ideal A in a compact semi-normal semigroup is generalized Archimedean iff A is contained in the intersection of all open prime ideals in S.
II. Let S be a compact semi-normal semigroup with \(S^ 2=S\) and \(M^*\) denote the intersection of all maximal ideals of S. Then \(M^*\) is generalized Archimedean iff \(M^*\) coincides with the intersection of all open prime ideals.
Reviewer: H.L.Vasudeva


22A15 Structure of topological semigroups
Full Text: EuDML


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