Endomorphisms of semigroups of oriented transformations

TL;DR

This paper characterizes the endomorphism monoids of various semigroups of oriented transformations on finite chains, providing explicit descriptions and counts of these endomorphisms.

Contribution

It offers new characterizations and enumeration of endomorphisms for semigroups of oriented transformations and their subsemigroups on finite chains.

Findings

Characterization of endomorphism monoids for all oriented transformations

Enumeration of endomorphisms for six semigroups

Descriptions of endomorphisms of orientation-preserving subsemigroups

Abstract

In this paper, we characterize the monoid of endomorphisms of the semigroup of all oriented full transformations of a finite chain, as well as the monoid of endomorphisms of the semigroup of all oriented partial transformations and the monoid of endomorphisms of the semigroup of all oriented partial permutations of a finite chain. Characterizations of the monoids of endomorphisms of the subsemigroups of all orientation-preserving transformations of the three semigroups aforementioned are also given. In addition, we compute the number of endomorphisms of each of these six semigroups.