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

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 $O(X)=⟨J⟩$ to hold. We also present a sufficient condition on $X$ for $O(X)=⟨J⟩$ to hold, for an arbitrary infinite chain $X$.

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• On Relative Ranks of the Semigroup of Orientation-preserving Transformations on Infinite Chains
• On orientation-preserving transformations of a chain