Geodesic
A note on intra regularity on semigroups of partial transformations with invariant set
Discussiones Mathematicae. General Algebra and Applications, Tome 44 (2024) no. 2, pp. 333-341

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Let X be any non-empty set and P(X) denote the semigroup (under composition) of partial transformations on a set X. Let Y be a fixed non-empty subset of X and PT(X,Y) = {α∈ P(X) : (domα∩ Y)α⊆ Y}. Then PT(X,Y) is a semigroup consisting of all mapping in P(X) that leave Y ⊆ X invariant. In this paper, we present criteria for checking the intra-regularity of elements in PT(X,Y) and apply these results to quantify intra-regular elements in PT(X,Y), when X is finite.
Keywords: partial transformation semigroup, intra regularity, invariant set 
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Pantarak, Thapakorn; Chaiya, Yanisa. A note on intra regularity on semigroups of partial transformations with invariant set. Discussiones Mathematicae. General Algebra and Applications, Tome 44 (2024) no. 2, pp. 333-341. http://geodesic.mathdoc.fr/item/DMGAA_2024_44_2_a6/