Abstract
It is well known that in a topological semigroup S with an identity 1 the maximal subgroup H (1) must be open if 1 has an euclidean neighborhood (Mostert-Shields [7]). If multiplication in S is only “separately” continuous, i.e. x↦yx and x↦xy is continuous for all y∈S, the statement remains true if the underlying space of S is a compact manifold (Berglund [2], Lawson [6]). In this paper the case of a compact semigroup with only one-sided continuity (i.e. x↦yx or x↦xy is continuous for all y∈S) which is defined on an interval or a circle is investigated. It is also shown that a group, defined on the line or on a circle, must be a topological group if it satisfies this very weak condition.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Literatur
BERGLUND, J.F.: Compact connected ordered semitopological semigroups. J. London math. Soc., II.Ser.,4, 533–540, (1972)
BERGLUND, J.F.: Semitopological semigroups on circles. J. London math. Soc., II. Ser.5, 395–398, (1972)
CLIFFORD, A.H., and PRESTON, G.B.: The algebraic theory of semigroups I. Math. Surveys 7 (Amer. Math. Soc., Providence, 1961)
ELLIS, R.: Locally compact transformation groups. Duke math. J.24, 119–126, (1957)
HOFMANN, K.H., and MOSTERT, P.S.: Elements of compact semigroups. (1. Aufl. Columbus, Ohio, C.E. Merrill, 1966)
LAWSON, J.D.: Joint continuity in semitopological semigroups, abstract in Amer.math.Soc., Notices20, (1973)
MOSTERT, P.S., and SHIELDS, A.L.: Semigroups with identity on a manifold. Trans.Amer.math.Soc.91, 380–389,(1959)
RUPPERT, W.: Rechtstopologische Halbgruppen. J. reine angew. Math.261, 123–133, (1973)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Ruppert, W. Rechtstopologische Intervallhalbgruppen und Kreishalbgruppen. Manuscripta Math 14, 183–193 (1974). https://doi.org/10.1007/BF01171441
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01171441