# On semigroups of endomorphisms of a chain with restricted range

Vítor H. Fernandes, Preeyanuch Honyam, Teresa M. Quinteiro, Boorapa Singha

Research output: Contribution to journalArticle

9 Citations (Scopus)

### Abstract

Let X be a finite or infinite chain and let ${\mathcal{O}}(X)$ be the monoid of all endomorphisms of X. In this paper, we describe the largest regular subsemigroup of ${\mathcal{O}}(X)$ and Green’s relations on ${\mathcal{O}}(X)$. In fact, more generally, if Y is a nonempty subset of X and ${\mathcal{O}}(X,Y)$ is the subsemigroup of ${\mathcal{O}}(X)$ of all elements with range contained in Y, we characterize the largest regular subsemigroup of ${\mathcal{O}}(X,Y)$ and Green’s relations on ${\mathcal{O}}(X,Y)$. Moreover, for finite chains, we determine when two semigroups of the type ${\mathcal {O}}(X,Y)$ are isomorphic and calculate their ranks.

Original language English 77-104 28 Semigroup Forum 89 1 https://doi.org/10.1007/s00233-013-9548-x Published - 1 Aug 2014

### Keywords

• Order-preserving
• Rank
• Restricted range
• Transformations