On semigroups of transformations that preserve a double direction equivalence
Hui Chen, Xin Liu, Shoufeng Wang · Open Mathematics · 2023
Abstract For a non-empty set X X , denote the full transformation semigroup on X X by T ( X ) T\left(X) and suppose that E E is an equivalence relation on X X . Evidently, T E ∗ ( X ) = { α ∈ T ( X ) ∣ ( x , y ) ∈ E if and only if ( x α , y α ) ∈ E for all x , 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} }} -abunda