Abstract
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 ) = P ( X ) \overline{PT}\left(X,X)=P\left(X) . In this article, we present the necessary and sufficient conditions for elements of P T ¯ ( X , Y ) \overline{PT}\left(X,Y) to be regular, left regular, and right regular. The results are used to describe the relationships between these elements and determine their number when X X is a finite set. Moreover, we show that P T ¯ ( X , Y ) \overline{PT}\left(X,Y) is always abundant.
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