The monoid of all partial transformations with restricted range

Let [Formula: see text] be a nonempty set and let [Formula: see text] be the monoid of all partial transformations of [Formula: see text]. Given a nonempty subset [Formula: see text] of [Formula: see text], we denote by [Formula: see text] the subsemigroup of [Formula: see text] of all partial transformations with range contained in [Formula: see text]. In 2014, Fernandes and Sanwong [On the ranks of semigroups of transformations on a finite set with restricted range, Algebra Colloquium. 21(3) (2014) 497–510] proved that the semigroup [Formula: see text] is the largest regular subsemigroup of [Formula: see text] and showed that this subsemigroup determines Green’s relations on [Formula: see text]. In this paper, we classify completely the maximal subsemigroups of [Formula: see text] when [Formula: see text]. Moreover, we prove that the rank of [Formula: see text] is equal to [Formula: see text] when [Formula: see text] and [Formula: see text].