Small doubling in ordered semigroups

Let $\mathbb{A} = (A, \cdot)$ be a semigroup. We generalize some recent results by G. A. Freiman, M. Herzog and coauthors on the structure theory of set addition from the context of linearly orderable groups to linearly orderable semigroups, where we say that $\mathbb{A}$ is linearly orderable if there exists a total order $\le$ on $A$ such that $xz < yz$ and $zx < zy$ for all $x,y,z \in A$ with $x < y$. In particular, we find that if $S$ is a finite subset of $A$ generating a non-abelian subsemigroup of $\mathbb{A}$, then $|S^2| \ge 3|S|-2$. On the road to this goal, we also prove a number of subsidiary results, and most notably that for $S$ a finite subset of $A$ the commutator and the normalizer of $S$ are equal to each other.