@article {2620, title = {On semigroups of endomorphisms of a chain with restricted range}, journal = {Semigroup Forum (DOI: 10.1007/s00233-013-9548-x)}, volume = {89}, year = {2014}, pages = {77-104}, 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{\textquoteright}s relations on $\O(X)$.
In fact, more generally, if $Y$ is a nonempty subset of $X$ and $\O(X,Y)$ is 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{\textquoteright}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.

}, url = {http://link.springer.com/article/10.1007\%2Fs00233-013-9548-x}, author = {Fernandes, V{\'\i}tor H. and Preeyanuch Honyam and Quinteiro, Teresa M. and Boorapa Singha} }