Regularity of the partial Baer-Levi semigroups with restricted range

Abstract

Let \(Y\) be a fixed nonempty subset of an infinite set \(X\) and let \(q\) be an infinite cardinal such that \(q\leq|X|\). Let \(PS(X,Y,q)\) denote the semigroup of all partial injective transformations from \(X\) into \(Y\) for which the complement of its range has cardinality \(q\). Then \(PS(X,Y,q)\) is a generalization of the partial Baer-Levi semigroup. In this paper, we study several types of regularity on \(PS(X, Y,q)\). We characterize all regular, left regular, right regular, completely regular, intra-regular and coregular elements and determine the largest regular subsemigroup of this semigroup. Furthermore, when \(Y\) is finite, we present formulas for counting the total number of elements of each type  mentioned above.