Is it correct that any ordered semigroup $S$ can be embedded into an ordered semigroup with zero $S_0$ in which every element is comparable with $0$, in a way that the order of $S$ is a subset of the order of $S_0$?
If not, is it always possible to re-order an ordered semigroup in this way?
The question was moved from here: Ordered semigroup with an absorbing element
The answer to the first question is yes.
Let $S$ be an ordered semigroup and let $S_0 = S \cup \{0\}$, where $0$ is a new zero element, that is, $0s = s0 = 0$ for all $s \in S_0$. Then the order on $S$ can be extended by adding the condition $0 < s$ for all $s \in S$.