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In this paper, we characterize the monoid of endomorphisms of the semigroup of all monotone full transformations of a finite chain, as well as the monoids of endomorphisms of the semigroup of all monotone partial transformations and of the semigroup of all monotone partial permutations of a finite chain.

Let [Formula: see text] be a finite chain and let [Formula: see text] be the semigroup of all injective order-preserving partial transformations on [Formula: see text]. For any nonempty subset [Formula: see text] of [Formula: see text], let [Formula: see text] be the subsemigroup of [Formula: see text] of all transformations with range contained in [Formula: see text]. In this paper, we characterize Green’s relations on [Formula: see text], and show that the semigroup [Formula: see text] is left abundant but not right abundant when [Formula: see text] is a proper subset of [Formula: see text]. Moreover, the cardinality and the rank of the semigroup [Formula: see text] is determined.

Let [Formula: see text] be the monoid of all orientation-preserving and extensive full transformations on [Formula: see text] ordered in the standard way. In this paper, we determine the minimum generating set and the minimum idempotent generating set of [Formula: see text], and so the rank and the idempotent rank of [Formula: see text] are obtained. Moreover, we describe maximal subsemigroups and maximal idempotent generated subsemigroups of [Formula: see text] and completely obtain their classifications.