Shum, Kar Ping Generalized Archimedean ideals. (English) Zbl 0579.22005 Czech. Math. J. 34(109), 604-608 (1984). 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 MSC: 22A15 Structure of topological semigroups Keywords:generalized Archimedean ideal; compact semigroup; semi-normal PDF BibTeX XML Cite \textit{K. P. Shum}, Czech. Math. J. 34(109), 604--608 (1984; Zbl 0579.22005) Full Text: EuDML References: [1] Fnip R.: Generalized semigroup kernels. Pacific Jour. Math. 24 (1968), p. 93-101. · Zbl 0165.03501 [2] Hofmann K. H., Mostert P. S.: Elements of compact semigroups. Charles E. Merrill Books, Inc., Columbus, Ohio) · Zbl 0161.01901 [3] Howie J. M.: An introduction to semigroup theory. Academic Press (1976). · Zbl 0355.20056 [4] Koch R. J, and Wallace A. D.: Maximal ideals in compact semigroups. Duke Math. Jour. 21 (1954), p. 681-686. · Zbl 0057.01502 [5] Numakura K.: Prime ideals and idempotents in compact semigroups. Duke Math. Jour. 24 (1957), p. 671-680. · Zbl 0218.22004 [6] Satyanarayana M.: On generalized kernels. Semigroup Forum 12 (1976), p. 283-292. · Zbl 0344.20048 [7] Schwarz Š.: Prime ideals and maximal ideals in semigroups. Czechoslovak Math. Jour. 24 (1968), p. 93-101. [8] Schwarz Š.: K teorii Chausdorfovych bikompaktnych polugrupp. Czechoslovak Math. Journ. 5 (1955), p. 1-23. [9] Shunt K. P., Hung C. Y.: Topological radicals in topological semigroups. Publ. Math. Debrecen, 29 (1982), p. 265-274. · Zbl 0508.22008 [10] Van der Waerden: Moderne Algebra. Springer-Verlag, Berlin, 1931. · Zbl 0002.00804 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.