Small doubling in ordered semigroups

Salvatore Tringali
2014 Semigroup Forum  
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 | ≥ 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
more » ... set 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. 2010 Mathematics Subject Classification. 06A07, 06F05, 20M10, 20N02. Net of a number of (minor) simplifications, our proof of Theorem 1 basically follows the same broad scheme as the proof of [6, Theorem 1.2]. However, the increased generality implied by the switching to the setting of linearly ordered semigroups raises a number of challenges and requires more than a mere adjustment of terminology, and especially the refinement of several classical results on linearly orderable groups, such as the following: Corollary 2. Let A = (A, ·, ) be a linearly ordered semigroup (written multiplicatively) and a, b ∈ A. If a n b = ba n for some n ∈ N + , then ab = ba. Corollary 2 is actually a generalization of an old lemma by N.H. Neumann [13] on commutators of linearly ordered groups, appearing as Lemma 2.2 in [6]; we prove it in Section 2.3. The next proposition is an extension of classical lower bounds on the size of product-sets of finite subsets of linearly ordered groups to the setting of linearly ordered magmas. Proposition 9. Suppose that A = (A, ·, ) is a linearly ordered magma (written multiplicatively). Pick n ∈ N + and let S 1 , S 2 , . . . , S n be nonempty finite subsets of A. Then (1) |(S 1 S 2 · · · S n ) P | ≥ 1 − n + n i=1 |S i | for any given parenthetization P of A of length n. The reader might want to consult [4] and the references therein for similar results in the context of arbitrary groups (notably including the Cauchy-Davenport theorem). Proposition 9 is proved in Section 2.3. Here, as is expected, we use ≥ (and its dual ≤) for the standard order of the real numbers (unless an explicit statement to the contrary) and, if S is a set, we denote by |S| the cardinality of S. More notation and terminology used in this introduction without explanation will be clarified below, in Section 2.1. We give two simple applications of Proposition 9, none of them covered by less general formulations of the same result such as the (classical) one reported in [6] for linearly ordered groups: The second one concerns the set of all upper (respectively, lower) triangular matrices with positive real entries, which we prove to be a linearly orderable semigroup (with respect to the usual matrix multiplication) in Proposition 1. In this respect, we raise the question, at present open to us, whether the same holds true for the set of all matrices which are a (finite) product of upper or lower triangular matrices with positive real entries. Then, we combine Proposition 9 with other basic properties to establish the following: Proposition 10. Let A = (A, ·, ) be a linearly ordered semigroup (written multiplicatively) and S a nonempty finite subset of A of size m, and pick y ∈ A \ C A (S). Then |yS ∪ Sy| ≥ m + 1, so in particular there exist a, b ∈ S such that ya / ∈ Sy and by / ∈ yS. Proposition 10 is a generalization of [6, Proposition 2.4]. We prove it in Section 3, along with the following interesting result, which in turn generalizes [6, Corollary 1.5]. Corollary 3. If S is a finite subset of a linearly ordered semigroup A, then N A (S) = C A (S). The whole is accompanied by a significant number of examples, mostly finalized to explore conditions under which some special classes of semigroups (or more sophisticated structures as semirings) are linearly orderable. In particular, we show by Proposition 6 that every abelian torsion-free cancellative semigroup is linearly orderable, so extending a similar 1913 result of F.W. Levi on abelian torsion-free groups.
doi:10.1007/s00233-014-9603-2 fatcat:pwt6726dk5aidgxqzmca7trrle