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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



The translational hull of a topological semigroup

Authors: J. A. Hildebrant, J. D. Lawson and D. P. Yeager
Journal: Trans. Amer. Math. Soc. 221 (1976), 251-280
MSC: Primary 22A15
MathSciNet review: 0409712
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Abstract: This paper is concerned with three aspects of the study of topological versions of the translational hull of a topological semigroup. These include topological properties, applications to the general theory of topological semigroups, and techniques for computing the translational hull. The central result of this paper is that if S is a compact reductive topological semigroup and its translational hull $ \Omega (S)$ is given the topology of continuous convergence (which coincides with the topology of pointwise convergence and the compact-open topology in this case), then $ \Omega (S)$ is again a compact topological semigroup. Results pertaining to extensions of bitranslations are given, and applications of these together with the central result to semigroup compactifications and divisibility are presented. Techniques for determining the translational hull of certain types of topological semigroups, along with numerous examples, are set forth in the final section.

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Keywords: Translational hull, topological semigroup, reductive semigroup, net reductive semigroup, bitranslation, topology of continuous convergence, topology of pointwise convergence, extension, semigroup compactification, uniquely divisible semigroup, basis, separating subset
Article copyright: © Copyright 1976 American Mathematical Society