Small doubling in ordered semigroups

Salvatore Tringali (1)

1 LJLL - Laboratoire Jacques-Louis Lions

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 $|S^2|\ge 3|S|-2$. 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.

Mots clés: Freiman's theory; ordered magmas; ordered semigroups; product-sets; small doubling; sum-sets; torsion-free semigroups

Domaines: Combinatoire [math.CO]; Théorie des groupes [math.GR]

Salvatore Tringali. Small doubling in ordered semigroups. 2012. ⟨hal-00723963v3⟩