On semigroups of transformations that preserve a double direction equivalence

Abstract

For a non-empty set XX, denote the full transformation semigroup on XX by T(X)T\left(X) and suppose that EE is an equivalence relation on XX. 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 of T(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 and G{\mathcal{G}}-abundant elements and left and right magnifying elements in TE∗(X){T}_{{E}^{\ast }}\left(X). More specifically, we first obtain the necessary and sufficient conditions under which ℒ{\mathcal{ {\mathcal L} }} (respectively, ℒ∗{{\mathcal{ {\mathcal L} }}}^{\ast }, ℒ˜\widetilde{{\mathcal{ {\mathcal L} }}}, ℛ{\mathcal{ {\mathcal R} }}, ℛ∗{{\mathcal{ {\mathcal R} }}}^{\ast }, and ℛ˜\widetilde{{\mathcal{ {\mathcal R} }}}) is (left, right) compatible, ℛ=ℛ∗{\mathcal{ {\mathcal R} }}={{\mathcal{ {\mathcal R} }}}^{\ast } or ℒ=ℒ˜{\mathcal{ {\mathcal L} }}=\widetilde{{\mathcal{ {\mathcal L} }}}. Then, we give the sufficient and necessary conditions such that TE∗(X){T}_{{E}^{\ast }}\left(X) is left regular (respectively, right regular, completely regular, intra-regular, and completely simple). Finally, we characterize the ℱ{\mathcal{ {\mathcal F} }}-abundant (respectively, G{\mathcal{G}}-abundant) and left (respectively, right) magnifying elements in TE∗(X){T}_{{E}^{\ast }}\left(X).

Keywords

• green (∼-, ∗-) relations
• (ℱ-, 𝒢-) abundant
• (left, right, intra-) regular
• (left, right) magnifying
• 20m20
• 20m10