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LetP(X)P\left(X)be a partial transformation semigroup on a non-empty setXX. For a fixed non-empty subsetYYofXX, letPT¯(X,Y)={α∈P(X)∣(domα∩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,PT¯(X,Y)\overline{PT}\left(X,Y)consists of all the mapping inP(X)P\left(X)that leaveY⊆XY\subseteq Xas an invariant. It is a generalization ofP(X)P\left(X)sincePT¯(X,X)=P(X)\overline{PT}\left(X,X)=P\left(X). In this article, we present the necessary and sufficient conditions for elements ofPT¯(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 whenXXis a finite set. Moreover, we show thatPT¯(X,Y)\overline{PT}\left(X,Y)is always abundant.

Regularity and abundance on semigroups of partial transformations with invariant set