University of Hertfordshire

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Holonomy Embedding for Arbitrary Stable Semigroups

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Original languageEnglish
Pages (from-to)791-810
JournalInternational Journal of Algebra and Computation
Volume12
Issue6
DOIs
Publication statusPublished - 2002

Abstract

We show how the Rhodes expansion Ŝ of any stable semigroup S embeds into the cascade integral (a natural generalization of the wreath product) of permutation-reset transformation semigroups with zero adjoined. The permutation groups involved are exactly the Schützenberger groups of the -classes of S. Since S ←← Ŝ is an aperiodic map via which all subgroups of S lift to Ŝ, this results in a strong Krohn–Rhodes–Zeiger decomposition for the entire class of stable semigroups. This class includes all semigroups that are finite, torsion, finite -above, compact Hausdorff, or relatively free profinite, as well as many other semigroups. Even if S is not stable, one can expand it using Henckell's expansion and then apply our embedding. This gives a simplified proof of the Holonomy Embedding theorem for all semigroups.

Notes

Original article can be found at: http://ejournals.wspc.com.sg/journals/ijac/mkt/archive.shtml Copyright World Scientific Publishing Company. DOI: 10.1142/S0218196702001206 [Full text of this article is not available in the UHRA]

ID: 97493