Regularity and abundance on semigroups of partial transformations with invariant set
Thapakorn Pantarak, Yanisa Chaiya · Open Mathematics · 2023
Abstract Let P ( X ) P\left(X) be a partial transformation semigroup on a non-empty set X X . For a fixed non-empty subset Y Y of X X , let P T ¯ ( X , Y ) = { α ∈ P ( X ) ∣ ( dom α ∩ Y ) α ⊆ Y } . \overline{PT}\left(X,Y)=\left\{\alpha \in P\left(X)| \left({\rm{dom}}\hspace{0.33em}\alpha \cap Y)\alpha \subseteq Y\right\}. Then, P T ¯ ( X , Y ) \overline{PT}\left(X,Y) consists of all the mapping in P ( X ) P\left(X) that leave Y ⊆ X Y\subseteq X as an invariant. It is a generalization of P ( X ) P\left(X) since P T ¯ ( X , X )