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

Araújo, João; Mitchell, James D.; Csaba, Schneider

http://hdl.handle.net/10400.2/3799

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Authors

Araújo, João

Mitchell, James D.

Csaba, Schneider

Abstract(s)

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

Keywords

Transformation semigroups Idempotent generated semigroups Regular semigroups Permutation groups Primitive groups OʼNan–Scott Theorem

URI

http://hdl.handle.net/10400.2/3799

Citation

Araújo, João; Mitchell, James D.; Schneider, Csaba - Groups that together with any transformation generate regular semigroups or idempotent generated semigroups. "Journal of Algebra" [Em linha]. ISSN 0021-8693. Vol. 343, nº 1 (2011), p. 1-16