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 $X$.

}, url = {http://dx.doi.org/10.1142/S0219498817500311 }, author = {Dimitrova, I. and Fernandes, V{\'\i}tor H. and Koppitz, J.} }