The regular part of transformation semigroups that preserve double direction equivalence relation

Abstract

Let $T(X)$ be the full transformation semigroup on a set $X$ under the composition of functions. For any equivalence relation $E$ on $X$, define a subsemigroup $T_{E^{\ast }}(X)$ of $T(X)$ by $$T_{E^{\ast }}(X)=\{\alpha \in T(X):\text{for all}x,y\in X,(x,y)\in E\iff (x\alpha ,y\alpha )\in E\}.$$ In this paper, we show that the regular part of $T_{E^{\ast }}(X)$, denoted $\mathrm{Reg}(T)$, is the largest regular subsemigroup of $T_{E^{\ast }}(X)$. Then its Green's relations and ideals are described. Moreover, we find the kernel of $\mathrm{Reg}(T)$ which is a right group and can be written as a union of symmetric groups. Finally, we prove that every right group can be embedded in that kernel.

12 pages, part of this work was presented in the Workshop on General Algebra / Arbeitstagung Allgemeine Algebra (AAA102), University of Szeged, Hungary, June 24-26, 2022