Narrow Results

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 $|\mathrm{\text{Im}}(\alpha )|=\left|X\right|$. 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 $X$.

In this paper, we determine the relative rank of the semigroup ${𝒪}{𝒫}(X)$ of all orientation-preserving transformations on infinite chains modulo the semigroup $Ø(X)$ of all order-preserving transformations.