Natural partial order on semigroups of partial transformations with invariant set

Abstract

Let \(X\) be a non-empty set, and let \(P(X)\) denote the semigroup of partial transformations on \(X\). For a non-empty subset \(Y\) of \(X\), define \(\overline{PT}(X,Y) = \{\alpha \in P(X) \mid (\text{dom } \alpha \cap Y)\alpha \subseteq Y \}.\) The semigroup \(\overline{PT}(X,Y)\) generalizes \(P(X)\) and consists of all partial transformations on \(X\) that leave \(Y\) invariant. In this paper, we investigate the natural partial order on \(\overline{PT}(X,Y)\) and characterize its left-compatible, right-compatible, minimal, and maximal elements. The results obtained extend and unify several known properties of \(P(X).\)