Small doubling in ordered semigroups

TL;DR

This paper extends the structure theory of set addition from linearly orderable groups to semigroups, establishing lower bounds on the size of product sets and exploring algebraic properties like commutators and normalizers.

Contribution

It generalizes recent additive combinatorics results to linearly orderable semigroups, providing new bounds and algebraic insights.

Findings

If S generates a non-abelian subsemigroup, then |S^2| ≥ 3|S| - 2.

The commutator and the normalizer of a finite subset S are equal.

Established lower bounds on product set sizes in ordered semigroups.

Abstract

Let $\mathbb{A} = (A, \cdot)$ be a semigroup. We generalize some recent results by G. A. Freiman, M. Herzog and coauthors on the structure theory of set addition from the context of linearly orderable groups to linearly orderable semigroups, where we say that $\mathbb{A}$ is linearly orderable if there exists a total order $\le$ on $A$ such that $xz < yz$ and $zx < zy$ for all $x,y,z \in A$ with $x < y$. In particular, we find that if $S$ is a finite subset of $A$ generating a non-abelian subsemigroup of $\mathbb{A}$, then $|S^2| \ge 3|S|-2$. On the road to this goal, we also prove a number of subsidiary results, and most notably that for $S$ a finite subset of $A$ the commutator and the normalizer of $S$ are equal to each other.