The translational hull of a topological semigroup
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- by J. A. Hildebrant, J. D. Lawson and D. P. Yeager PDF
- Trans. Amer. Math. Soc. 221 (1976), 251-280 Request permission
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.References
- Richard Arens and James Dugundji, Topologies for function spaces, Pacific J. Math. 1 (1951), 5–31. MR 43447 T. T. Bowman, The semilattice of left translations of a compact semilattice, Notices Amer. Math. Soc. 20 (1973), p. A-673. Abstract #709-G16.
- Francis T. Christoph Jr., Ideal extensions of topological semigroups, Canadian J. Math. 22 (1970), 1168–1175. MR 268317, DOI 10.4153/CJM-1970-134-0
- A. H. Clifford and G. B. Preston, The algebraic theory of semigroups. Vol. I, Mathematical Surveys, No. 7, American Mathematical Society, Providence, R.I., 1961. MR 0132791
- Michael Friedberg, On compactifying semigroups, Semigroup Forum 10 (1975/76), no. 1, 39–45. MR 384980, DOI 10.1007/BF02194871
- J. A. Hildebrant and J. D. Lawson, Embedding in compact uniquely divisible semigroups, Semigroup Forum 4 (1972), 295–300. MR 304516, DOI 10.1007/BF02570801
- J. A. Hildebrant and J. D. Lawson, The Bohr compactification of a dense ideal in a topological semigroup, Semigroup Forum 6 (1973), no. 1, 86–92. MR 401979, DOI 10.1007/BF02389112
- Karl Heinrich Hofmann and Paul S. Mostert, Elements of compact semigroups, Charles E. Merrill Books, Inc., Columbus, Ohio, 1966. MR 0209387
- John L. Kelley, General topology, D. Van Nostrand Co., Inc., Toronto-New York-London, 1955. MR 0070144
- J. D. Lawson, Joint continuity in semitopological semigroups, Illinois J. Math. 18 (1974), 275–285. MR 335674
- Mario Petrich, The translational hull in semigroups and rings, Semigroup Forum 1 (1970), no. 4, 283–360. MR 267017, DOI 10.1007/BF02573051
- Mario Petrich, Introduction to semigroups, Merrill Research and Lecture Series, Charles E. Merrill Publishing Co., Columbus, Ohio, 1973. MR 0393206
Additional Information
- © Copyright 1976 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 221 (1976), 251-280
- MSC: Primary 22A15
- DOI: https://doi.org/10.1090/S0002-9947-1976-0409712-8
- MathSciNet review: 0409712