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Transactions of the American Mathematical Society

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



All varieties of central completely simple semigroups

Authors: Mario Petrich and Norman R. Reilly
Journal: Trans. Amer. Math. Soc. 280 (1983), 623-636
MSC: Primary 20M07
MathSciNet review: 716841
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Abstract: Completely simple semigroups may be considered as a variety of algebras with the binary operation of multiplication and the unary operation of inversion. A completely simple semigroup is central if the product of any two idempotents lies in the centre of the containing maximal subgroup. Central completely simple semigroups form a subvariety $ \mathcal{C}$ of the variety of all completely simple semigroups. We find an isomorphic copy of $ \mathcal{L}(\mathcal{C})$ as a subdirect product of the lattices $ \mathcal{L}(\mathcal{R}\,\mathcal{B})$, $ \mathcal{L}(\mathcal{A}\,\mathcal{G})$, and $ \mathcal{L}(\mathcal{G})$ of all varieties of rectangular bands, abelian groups, and groups, respectively. We consider also several homomorphisms and study congruences they induce.

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  • [1] A. H. Clifford, The free completely regular semigroup on a set, J. Algebra 59 (1979), 434-451. MR 543262 (80h:20082a)
  • [2] J. M. Howie, An introduction to semigroup theory, Academic Press, London, 1976. MR 0466355 (57:6235)
  • [3] J. Leech, The structure of a band of groups, Mem. Amer. Math. Soc. 1 (1975), No. 157, pp. 67-95. MR 0372084 (51:8301)
  • [4] G. I. Maševickiĭ, On identities in varieties of completely simple semigroups over abelian groups, Contemporary Algebra, Leningrad, (1978), 81-89. (Russian)
  • [5] M. Petrich, Introduction to semigroups, Merrill, Columbus, Ohio, 1973. MR 0393206 (52:14016)
  • [6] M. Petrich and N. R. Reilly, Varieties of groups and of completely simple semigroups, Bull. Austral. Math. Soc. 23 (1981), 339-359. MR 625177 (82j:20107)
  • [7] -, Near varieties of idempotent generated completely simple semigroups, Algebra Universalis (to appear). MR 690832 (84i:20061)
  • [8] V. V. Rasin, On the lattice of varieties of completely simple semigroups, Semigroup Forum 17 (1979), 113-122. MR 527213 (80e:20073)
  • [9] -, Free completely simple semigroups, Mat. Zapiski Ural. Univ. 11 (1979), 140-151. (Russian) MR 573889 (81i:20079a)

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Keywords: Completely simple semigroups, varieties, central completely simple semigroups, fully invariant normal subgroups, free completely simple semigroups
Article copyright: © Copyright 1983 American Mathematical Society

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