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A generalization of the rees theorem to a class of regular semigroups

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Abstract

Let S be a regular semigroup for which Green's relations J and D coincide, and which is max-principal in the sense that every element of S is contained in maximal principal right, left and two-sided ideals of S. A construction is given of a max-principal regular semigroup W with J=D, which is also principally separated in the sense that distinct maximal principal right (or left) ideals of S are disjoint, and an epimorphism ψ: W→S that preserves maximality of principal left, right, and two sided ideals, and is in a sense locally one-to-one. If S is completely simple, this construction reduces to the Rees matrix representation of S.

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References

  1. ALLEN, D. Jr.: A structure theory for finite regular semigroups, to appear.

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Authors and Affiliations

  1. Michigan Technological University, 49931, Houghton, Michigan, U.S.A.

    Dennis Allen Jr.

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  1. Dennis Allen Jr.
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Additional information

Communicated by A.H. Clifford

The main result of this paper has its origin in an incorrect result contained in the author's doctoral dissertation which was written at the University of California (Berkeley) under Professor John Rhodes. This theorem was first established for finite regular semigroups in [1] (Corollary 2.3), and the present generalization of this result to infinite semigroups was suggested by Professor A. H. Clifford, who the author would like to thank for this as well as his generous encouragement and many helpful editorial suggestions. The author would also like to thank Professor Rhodes for his encouragement.

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Allen, D. A generalization of the rees theorem to a class of regular semigroups. Semigroup Forum 2, 321–331 (1971). https://doi.org/10.1007/BF02572298

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  • DOI: https://doi.org/10.1007/BF02572298

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