We study the structure of the ideals of the semigroup IO n of all isotone (order-preserving) partial injections as well as of the semigroup IM n of all monotone (order-preserving or order-reversing) partial injections on an n-element set. The main result is the characterization of the maximal subsemigroups of the ideals of IO n and IM n .

Let Singn be the semigroup of all singular transformations on an n-element set. We consider two subsemigroups of Singn: the semigroup On of all isotone singular transformations and the semigroup Mn of all monotone singular transformations. We describe the maximal subsemigroups of these two semigroups, and study the connections between them.

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For [Formula: see text], let [Formula: see text] be an [Formula: see text]-element set and let [Formula: see text] be a fence, also called a zigzag poset. As usual, we denote by [Formula: see text] the symmetric inverse semigroup on [Formula: see text]. We say that a transformation [Formula: see text] is fence-preserving if [Formula: see text] implies that [Formula: see text], for all [Formula: see text] in the domain of [Formula: see text]. In this paper, we study the semigroup [Formula: see text] of all partial fence-preserving injections of [Formula: see text] and its subsemigroup [Formula: see text]. Clearly, [Formula: see text] is an inverse semigroup and contains all regular elements of [Formula: see text] We characterize the Green’s relations for the semigroup [Formula: see text]. Further, we prove that the semigroup [Formula: see text] is generated by its elements with rank [Formula: see text]. Moreover, for [Formula: see text], we find the least generating set and calculate the rank of [Formula: see text].

The study of the semigroups OP n and OR n respectively of all orientation-preserving transformations and of all orientation-preserving or orientation-reversing transformations on an n-element chain has began in [10] and [4]. In order to bring more insight into the subsemigroup structure of OP n and OR n , we characterize their maximal subsemigroups.