On semigroups of endomorphisms of a chain with restricted range

Abstract

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.

On relative ranks of the semigroup of orientation-preserving transformations on infinite chains

“…For a finite chain X the ranks of the semigroups O(X, Y ) and OP(X, Y ) were determined by Fernandes, Honyam, Quinteiro, and Singha [6,7]. In [18], Tinpun and Koppitz studied the relative rank of T (X, Y ) modulo O(X, Y ).…”

Regularity of semigroups of transformations with restricted range preserving an alternating orientation order

“…Keprasit and Changphas [9] showed that if X is order-isomorphic to a subchain of Z, then OT (X) is regular. In [4], Fernandes et al described the largest regular subsemigroup of OT (X).…”

“…For a chain X , Mora and Kemprasit [14] gave a necessary and sufficient condition for OT (X, Y ) to be regular and determined all regular elements. Fernandes et al [4] characterized the largest regular subsemigroup of OT (X, Y ) .…”

A note on generators of the endomorphism semigroup of an infinite countable chain

“…Notice that J f is clearly an ideal of O(X). On the other hand, unlike the analogous set for T (X), J is not necessarily a J -class of O(X) (see [3]).…”