  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.
