%0 Journal Article %J Semigroup Forum (DOI: 10.1007/s00233-013-9548-x) %D 2014 %T On semigroups of endomorphisms of a chain with restricted range %A Fernandes, VĂ­tor H. %A Preeyanuch Honyam %A Quinteiro, Teresa M. %A Boorapa Singha %N 1 %P 77-104 %U http://link.springer.com/article/10.1007%2Fs00233-013-9548-x %V 89 %X

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 $\O(X,Y)$ 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'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.