On semigroups of endomorphisms of a chain with restricted range

• Main Authors: Vítor H. Fernandes, Preeyanuch Honyam, Teresa M. Quinteiro, Boorapa Singha
• Format: Journal
• Published: 2018
• Subjects: Mathematics
• Online Access: https://www.scopus.com/inward/record.uri?partnerID=HzOxMe3b&scp=84942193328&origin=inward; http://cmuir.cmu.ac.th/jspui/handle/6653943832/53664
• Institution: Chiang Mai University

© 2013, Springer Science+Business Media New York. 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.