Regular centralizers of idempotent transformations

André, Jorge; Araújo, João; Konieczny, Janusz

http://hdl.handle.net/10400.2/2001

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Authors

André, Jorge

Araújo, João

Konieczny, Janusz

Abstract(s)

Denote by T(X) the semigroup of full transformations on a set X. For ε∈T(X), the centralizer of ε is a subsemigroup of T(X) defined by C(ε)={α∈T(X):αε=εα}. It is well known that C(id X )=T(X) is a regular semigroup. By a theorem proved by J.M. Howie in 1966, we know that if X is finite, then the subsemigroup generated by the idempotents of C(id X ) contains all non-invertible transformations in C(id X ). This paper generalizes this result to C(ε), an arbitrary regular centralizer of an idempotent transformation ε∈T(X), by describing the subsemigroup generated by the idempotents of C(ε). As a corollary we obtain that the subsemigroup generated by the idempotents of a regular C(ε) contains all non-invertible transformations in C(ε) if and only if ε is the identity or a constant transformation.

Keywords

Idempotent transformations Regular centralizers Generators

URI

http://hdl.handle.net/10400.2/2001

Citation

André, Jorge; Araújo, João; Konieczny, Janusz - Regular centralizers of idempotent transformations. "Semigroup Forum" [Em linha]. ISSN 0037-1912 (Print) 1432-2137 (Online). Vol. 82, nº 2 (Apr. 2011), p. 307-318