Abstract
Given a nonempty set \(X\) and \(Y\subseteq X\), we denote by \(T(X,Y)\) the subsemigroup of the full transformation semigroup \(\mathcal{T}_X\) on \(X\) consisting of all transformations whose range is contained in \(Y\). In this paper, we endow the semigroup \(T(X,Y)\) with the natural partial order and investigate when two elements are related, then find elements which are compatible with the order. Also, we characterize the minimal elements and the maximal elements.
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