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Transactions of the American Mathematical Society

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Semigroups with a dense subgroup


Author: W. S. Owen
Journal: Trans. Amer. Math. Soc. 212 (1975), 219-228
MSC: Primary 22A15
DOI: https://doi.org/10.1090/S0002-9947-1975-0412330-8
MathSciNet review: 0412330
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Abstract: The purpose of this paper is twofold. First, it is shown that the ideal structure of a semigroup with dense subgroup is closely related to its transformation group structure. That is, if a left orbit through a given point is locally compact, then the members of this orbit are precisely those elements which generate the same left ideal as the given point.

Secondly, the author gives a number of theorems which have as their goal the establishment of a natural product structure near a nonzero idempotent. Specifically the work of F. Knowles [11] is improved upon to include (1) the possibility of a nonconnected group; (2) the possibility of a nonsimply connected orbit; and (3) the case in which the boundary of the group is more than a single orbit.


References [Enhancements On Off] (What's this?)

  • [1] A. H. Clifford and G. B. Preston, The algebraic theory of semigroups. Vols. I, II, Math. Surveys, no. 7, Amer. Math. Soc., Providence, R. I., 1961, 1967. MR 24 #A2627; 36 #1558. MR 0132791 (24:A2627)
  • [2] J. Dugundji, Topology, Allyn and Bacon, Boston, Mass., 1966. MR 33 #1824. MR 0193606 (33:1824)
  • [3] R. Ellis, A note on the continuity of the inverse, Proc. Amer. Math. Soc. 8 (1957), 372-373. MR 18, 745. MR 0083681 (18:745d)
  • [4] G. Hochschild, The structure of Lie groups, Holden-Day, San Francisco, Calif., 1965. MR 34 #7696. MR 0207883 (34:7696)
  • [5] J. G. Horne, Jr., Semigroups on a half-space, Trans. Amer. Math. Soc. 147 (1970), 1-53. MR 41 #378. MR 0255718 (41:378)
  • [6] F. Knowles, Semigroups that are the union of a group on $ {E^3}$ and a plane, Trans. Amer. Math. Soc. 160 (1971), 305-325. MR 43 #7545. MR 0281831 (43:7545)
  • [7] -, Idempotents in the boundary of a Lie group, Pacific J. Math. 44 (1973), 191-200. MR 47 #3592. MR 0315043 (47:3592)
  • [8] P. S. Mostert and A. L. Shields, Semigroups with identity on a manifold, Trans. Amer. Math. Soc. 91 (1959), 380-389. MR 21 #4204. MR 0105463 (21:4204)
  • [9] G. D. Mostow, The extensibility of local Lie groups of transformations and groups on surfaces, Ann. of Math. (2) 52 (1950), 606-636. MR 14, 18. MR 0048464 (14:18d)
  • [10] W. S. Owen, The Rees theorem for locally compact semigroups, Semigroup Forum 6 (1973), 133-152. MR 0348028 (50:526)
  • [11] N. E. Steenrod, The topology of fibre bundles, Princeton Math. Ser., vol. 14, Princeton Univ. Press, Princeton, N. J., 1951. MR 12, 522. MR 0039258 (12:522b)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1975-0412330-8
Keywords: Completely semisimple semigroup, dense Lie subgroup, Green's relations, local cross section, locally compact orbit
Article copyright: © Copyright 1975 American Mathematical Society

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