Groups that together with any transformation generate regular semigroups or idempotent generated semigroups

2009/11/02 by Joao Araujo, Araujo, Joao, J. D. Mitchell +3
Mathematics · #20B15 #20B30 #20B35 #20B40 #20M17 #20M20 #FOS: Mathematics #Group Theory (math.GR) #math.GR #msc:20B15 #msc:20B30 #msc:20B35 #msc:20B40 #msc:20M17 #msc:20M20

paper · pdf · doi:10.48550/arxiv.0911.0445

arxiv created 2009/11/02 · arxiv updated 2009/12/01

Abstract

Let a be a non-invertible transformation of a finite set and let G be a group of permutations on that same set. Then \gensetG, a∖ G is a subsemigroup, consisting of all non-invertible transformations, in the semigroup generated by G and a. Likewise, the conjugates a^g=g^-1ag of a by elements g∈ G generate a semigroup denoted \genseta^g | g∈ G. We classify the finite permutation groups G on a finite set X such that the semigroups \gensetG,a, \gensetG, a∖ G, and \genseta^g | g∈ G are regular for all transformations of X. We also classify the permutation groups G on a finite set X such that the semigroups \gensetG, a∖ G and \genseta^g | g∈ G are generated by their idempotents for all non-invertible transformations of X.

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