Small doubling in ordered semigroups
Résumé
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.
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Salvatore_Tringali-Small_doubling_in_ordered_semigroups.pdf (373.22 Ko)
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