Miscellaneous results on sum-sets in ordered semigroups and magmas

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 set addition from the context of linearly (i.e., strictly and totally) ordered groups (LOGs) to the one of linearly ordered semigroups (LOSs). In particular, we find that, in a LOS, the commutator and the normalizer of a finite set are equal to each other. On the road to this goal, we also extend an old lemma of B.H. Neumann on commutators of LOGs to the setting of LOSs and classical lower bounds on the size of sum-sets of finite subsets of LOGs to linearly ordered magmas. The whole is accompanied by a number of examples, one of these including a proof that the multiplicative semigroup of all upper (respectively, lower) triangular matrices with positive real entries is linearly orderable.

Mots clés: Sum-sets; product-sets; ordered magmas; ordered semigroups; Freiman's theory

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

Salvatore Tringali. Miscellaneous results on sum-sets in ordered semigroups and magmas. 2012. ⟨hal-00723963v1⟩