On the semigroup of injective transformations with restricted range that equal gap and defect

Abstract

Let \(X\) be an infinite set and \(I(X)\) the symmetric inverse semigroup on \(X\). Let \(A(X)=\{\alpha \in I(X):|X\setminus \mathrm{dom\;}\alpha|=|X\setminus X\alpha|\}\), it is known that \(A(X)\) is the largest factorizable subsemigroup of \(I(X)\). In this article, for any nonempty subset \(Y\) of \(X\), we consider the subsemigroup \(A(X, Y)\) of \(A(X)\) of all transformations with range contained in \(Y\). We give a complete description of Green's relations on \(A(X,Y)\). With respect to the natural partial order on a semigroup, we determine when two elements in \(A(X,Y)\) are related and find all the maximum, minimum, maximal, minimal, lower cover and upper cover elements. We also describe elements which are compatible and we investigate the greatest lower bound and the least upper bound of two elements in \(A(X,Y)\).