Small doubling in ordered semigroups

We generalize recent results by G.A. Freiman, M. Herzog and coauthors on the structure theory of product-sets from the context of linearly (i.e., strictly and totally) ordered groups to linearly ordered semigroups. In particular, we find that if S is a finite subset of a linearly ordered semigroup generating a nonabelian subsemigroup, then |S2| ≥ 3 |S| − 2. On the road to this goal, we also prove a number of subsidiary results, and notably that the commutator and the normalizer of a finite subset of a linearly ordered semigroup are equal to each other. The whole is accompanied by several examples, including a proof that the multiplicative semigroup of upper (respectively, lower) triangular matrices with positive real entries is linearly orderable.