序半群中的小加倍
Small doubling in ordered semigroups
摘要：
设$\mathbb{A} = (A, \cdot)$是半群.推广了G. A. Freiman，M. Herzog和合著者的结构理论集除了从上下文的线性序群线性序半群，我们说，$\mathbb{A}$是线性序，如果存在一个全序$\le$对$A$，使$xz < yz$和$zx < zy$的所有$x,y,z \in A$与$x < y$。特别地，我们发现如果$S$是生成$\mathbb{A}$的一个非交换子半群的$A$的有限子集，则$|S^2| \ge 3|S|-2$ .在通往这一目标的道路上，我们还证明了一些附属的结果，最值得注意的是，对于$S$有限子集的$A$的换向器和规范化的$S$是彼此相等的。
英文摘要：
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.
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