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arXiv · 1410.5253

Variants of finite full transformation semigroups

Igor Dolinka·James East
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Abstract

The variant of a semigroup S with respect to an element a in S, denoted S^a, is the semigroup with underlying set S and operation * defined by x*y=xay for x,y in S. In this article, we study variants T_X^a of the full transformation semigroup T_X on a finite set X. We explore the structure of T_X^a as well as its subsemigroups Reg(T_X^a) (consisting of all regular elements) and E_X^a (consisting of all products of idempotents), and the ideals of Reg(T_X^a). Among other results, we calculate the rank and idempotent rank (if applicable) of each semigroup, and (where possible) the number of (idempotent) generating sets of the minimal possible size.

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Igor Dolinka, James East. 2015-09-23. Variants of finite full transformation semigroups. https://doi.org/10.1142/s021819671550037x

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