A note on generators of the endomorphism semigroup of an infinite countable chain

TL;DR

This paper investigates the generators of the semigroup of all order endomorphisms of an infinite chain, providing conditions under which the entire semigroup is generated by transformations with large images.

Contribution

It establishes necessary and sufficient conditions for when the endomorphism semigroup of an infinite countable chain is generated by transformations with large images.

Findings

Characterizes when $O(X)$ equals the semigroup generated by $J$ for countable chains.

Provides sufficient conditions for arbitrary infinite chains.

Advances understanding of the algebraic structure of endomorphism semigroups of chains.

Abstract

In this note, we consider the semigroup $O(X)$ of all order endomorphisms of an infinite chain $X$ and the subset $J$ of $O(X)$ of all transformations $\alpha$ such that $|Im(\alpha)|=|X|$. For an infinite countable chain $X$, we give a necessary and sufficient condition on $X$ for $O(X) = \langle J \rangle$ to hold. We also present a sufficient condition on $X$ for $O(X) = \langle J \rangle$ to hold, for an arbitrary infinite chain $X$.