On semigroups of transformations that preserve a double direction equivalence

Abstract

AbstractFor a non-empty setXX, denote the full transformation semigroup onXXbyT(X)T\left(X)and suppose thatEEis an equivalence relation onXX. Evidently,TE∗(X)={α∈T(X)∣(x,y)∈Eif and only if(xα,yα)∈Efor allx,y∈X}{T}_{{E}^{\ast }}\left(X)=\left\{\alpha \in T\left(X)| \left(x,y)\in E\hspace{0.33em}\hspace{0.1em}\text{if and only if}\hspace{0.1em}\hspace{0.33em}\left(x\alpha ,y\alpha )\in E\hspace{0.33em}\hspace{0.1em}\text{for all}\hspace{0.1em}\hspace{0.33em}x,y\in X\right\}is a subsemigroup ofT(X)T\left(X). In this article, we investigate Green relations, Green∗\ast-relations and Green∼\sim-relations, various kinds of regularities,ℱ{\mathcal{ {\mathcal F} }}-abundant andG{\mathcal{G}}-abundant elements and left and right magnifying elements inTE∗(X){T}_{{E}^{\ast }}\left(X). More specifically, we first obtain the necessary and sufficient conditions under whichℒ{\mathcal{ {\mathcal L} }}(respectively,ℒ∗{{\mathcal{ {\mathcal L} }}}^{\ast },ℒ˜\widetilde{{\mathcal{ {\mathcal L} }}},ℛ{\mathcal{ {\mathcal R} }},ℛ∗{{\mathcal{ {\mathcal R} }}}^{\ast }, andℛ˜\widetilde{{\mathcal{ {\mathcal R} }}}) is (left, right) compatible,ℛ=ℛ∗{\mathcal{ {\mathcal R} }}={{\mathcal{ {\mathcal R} }}}^{\ast }orℒ=ℒ˜{\mathcal{ {\mathcal L} }}=\widetilde{{\mathcal{ {\mathcal L} }}}. Then, we give the sufficient and necessary conditions such thatTE∗(X){T}_{{E}^{\ast }}\left(X)is left regular (respectively, right regular, completely regular, intra-regular, and completely simple). Finally, we characterize theℱ{\mathcal{ {\mathcal F} }}-abundant (respectively,G{\mathcal{G}}-abundant) and left (respectively, right) magnifying elements inTE∗(X){T}_{{E}^{\ast }}\left(X).