Abstract
A regular semigroup S is called an ℋ-coextension of a regular semigroup T if there exists an idempotent-separating homomorhism from S onto T. J. Meakin [5] has described all regular four-spiral semigroups, i.e. all ℋ-coextensions of the fundamental four-spiral semigroup Sp4 [2], by means of the structure mappings on a regular semigroup. The purpose of this note is to point out that D. Allen's generalization [1] of the Rees theorem allows one to give a short alternative description of all regular four-spiral semigroups and their maximum completely simple homomorphic images in terms of bisimple ω-semigroups (whose structure is known by Reilly's theorem [7]) and Rees matrix semigroups ℳ(S;I;ΛP) over a semigroup S [3]. The notion of a Rees matrix semigroup over a semigroup is also used to embed semigroups in idempotent-generated ones, providing easy proofs for some embedding theorems of F. Pastijn [6].
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Allen, D.,A generalization of, the Rees theorem to a class of regular semigroups, Semigroup Forum 2 (1971), 321–331.
Byleen, K., J. Meakin and F. Pastijn,The fundamental fourspiral semigroup, J. Algebra 54 (1978), 6–26.
Eilenberg, S.,Automata, languages and machines, Vol. B, Academic Press, New York (1976), Chapters XI and XII by Bret Tilson.
Lallement, G.,Some remarks on the four-spiral semigroup, Semigroup Forum 18 (1979), 341–345.
Meakin, J.,Structure mappings, coextensions and regular four-spiral semigroups, Trans. Am. Math. Soc. 225 (1979), 111–134.
Pastijn, F.,Embedding semigroups in semibands Semigroup Forum 14 (1977), 247–263.
Reilly, N.,Bisimple ω-semigroups, Proc. Glasgow Math. Assoc. 7 (1966), 160–167.
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Communicated by D.B. McAlister
This research was supported by NSF Grant MCS78-00414-01.
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Byleen, K. Regular four-spiral semigroups, idempotent-generated semigroups and the rees construction. Semigroup Forum 22, 97–100 (1981). https://doi.org/10.1007/BF02572789
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DOI: https://doi.org/10.1007/BF02572789