Abstract
Some additional properties of the intersection of the maximal ideals of a compact semigroup are developed here based on results in [1] and [3]. Throughout S denotes a compact usually connected semigroup with at least one maximal proper ideal. The set of all such maximal ideals is denoted by ℳ, the intersection of the members of ℳ by R, the idempotents by E and the minimal ideal by K. Some proofs are more algebraic and in a few cases we do not need S connected. Key facts are that members of ℳ are open and dense, complements of distinct maximal ideals are disjoint and the union of any two such is S. After some generalization of results in [1] and [3], we investigate R relative to the topology of S. Necessary and sufficient conditions are found for R to be compact hence closed and for R to be open. Unlike the situation with the minimal ideal, R can be closed or open largely depending on the position of E relative to R. The following theorem summarizes necessary preliminaries from [1] and [3].
Similar content being viewed by others
References
Faucett, W. M., R. J. Koch and K. Numakura, Complements of maximal ideals in compact semigroups, Duke Math. J. 22(1955), 655–661.
Hofmann, K. and P. Mostert, Elements of Compact Semigroups. Charles E. Merrill Books, Columbus, Ohio, 1966.
Koch, R. J. and A. D. Wallace, Maximal ideals in compact semigroups, Duke Math. J. 21(1954), 681–686.
Koch, R. J. Remarks on primitive idempotents in compact semigroups with zero, Proc. Amer. Math. Soc. 5(1954), 828–833.
Author information
Authors and Affiliations
Additional information
Communicated by P. S. Mostert
Adapted from material in Chapter 3 of the author's dissertation, written under the co-direction of Dr. John Mack and Dr. John Selden at the University of Kentucky and supported by the National Science Foundation. Support to organize and prepare this paper was provided by Mount Vernon Nazarene College.
Rights and permissions
About this article
Cite this article
Dobbins, C. On the intersection of the maximal ideals. Semigroup Forum 22, 119–123 (1981). https://doi.org/10.1007/BF02572791
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF02572791