Un lemme combinatoire de H. B. Neumann

TL;DR

This paper presents a simpler, more concise proof of H. B. Neumann's theorem on the well-ordering of finite products in well-ordered subsets of totally ordered semigroups, highlighting the finiteness of product representations.

Contribution

It provides a notably simpler and shorter proof of Neumann's result on the structure of finite products in well-ordered semigroups.

Findings

Finite products of well-ordered subsets are well-ordered.

Only finitely many products equal a given element t.

Simplified proof enhances understanding of semigroup order properties.

Abstract

We give a notably simpler and shorter proof of H. B. Neumann's result which is stated, cursorly, like this. For any well-ordered subset, A, of a totally ordered semigroup, the set of products of any finite number of elements of A is itself well-ordered. Moreover, for each t, there are only a finite number of such products equal to t.