Let X be a finite or infinite chain and let O(X) be the monoid of all endomorphisms of X. In this paper, we describe the largest regular subsemigroup of O(X) and Green's relations on O(X). In fact, more generally, if Y is a nonempty subset of X and O(X, Y ) the subsemigroup of O(X) of all elements with range contained in Y , we characterize the largest regular subsemigroup of O(X, Y ) and Green's relations on O(X, Y ). Moreover, for finite chains, we determine when two semigroups of the type O(X, Y ) are isomorphic and calculate their ranks.2000 Mathematics subject classification: 20M20, 20M10.

Let Ω n be a finite chain with n elements (n ∈ N), and let POPI n be the semigroup of all injective orientation-preserving partial transformations of Ω n . In this paper, for any nonempty subset Y of Ω n , we consider the subsemigroup POPI n (Y ) of POPI n of all transformations with range contained in Y . We describe the Green's relations and study the regularity of POPI n (Y ). Moreover, we calculate the rank of POPI n (Y ) and determine when two semigroups of this type are isomorphic. 2020 Mathematics subject classification: 20M20, 20M10

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Marques-Smith and Sullivan ['Partial orders on transformation semigroups ', Monatsh. Math. 140 (2003), 103-118] studied various properties of two partial orders on P(X ), the semigroup (under composition) consisting of all partial transformations of an arbitrary set X . One partial order was the 'containment order': namely, if α, β ∈ P(X ) then α ⊆ β means xα = xβ for all x ∈ dom α, the domain of α. The other order was the so-called 'natural order' defined by Mitsch ['A natural partial order for semigroups', Proc. Amer. Math. Soc. 97(3) (1986), 384-388] for any semigroup. In this paper, we consider these and other orders defined on the symmetric inverse semigroup I (X ) and the partial Baer-Levi semigroup P S(q). We show that there are surprising differences between the orders on these semigroups, concerned with their compatibility with respect to composition and the existence of maximal and minimal elements.2000 Mathematics subject classification: primary 20M20; secondary 04A05, 06A06.

In this paper we introduce the notion of an orientation-preserving transformation on an arbitrary chain, as a natural extension for infinite chains of the well known concept for finite chains introduced in 1998 by McAlister [27] and, independently, in 1999 by Catarino and Higgins [8]. We consider the monoid POP(X) of all orientation-preserving partial transformations on a finite or infinite chain X and its submonoids OP(X) and POPI(X) of all orientation-preserving full transformations and of all orientation-preserving partial permutations on X, respectively. The monoid PO(X) of all order-preserving partial transformations on X and its injective counterpart POI(X) are also considered. We study the regularity and give descriptions of the Green's relations of the monoids POP(X), PO(X), OP(X), POPI(X) and POI(X).