Regularity and abundance on semigroups of partial transformations with invariant set

Let P(X) be a partial transformation semigroup on a non-empty set X . For a fixed non-empty subset Y of X , let PT(X,Y)={alpha is an element of P(X)divided by(dom alpha boolean AND Y)alpha subset of Y}.Then, PT(X,Y) consists of all the mapping in P(X) that leave Y subset of X as an invariant. It is a generalization of P(X) since PT(X,X)=P(X). In this article, we present the necessary and sufficient conditions for elements of PT(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 is a finite set. Moreover, we show that PT(X,Y) is always abundant.

Automated summary

In this article, we investigate the properties of the partial transformation semigroup PT(X,Y) and provide the necessary and sufficient conditions for its elements to be regular, left regular, and right regular. We also describe the relationships between these elements and determine their number in the case of a finite set X. Additionally, we demonstrate that PT(X,Y) is always abundant.