On Bounding Problems on Totally Ordered Commutative Semi-Groups

TL;DR

This paper proves bounds on pairwise products in totally ordered commutative semi-groups, establishing conditions under which certain pairs are guaranteed to be less than a given element, with applications to bounding problems.

Contribution

It introduces a new bounding principle for pairwise products in totally ordered semi-groups, extending previous results to both upper and lower bounds.

Findings

If a system of pairs has products less than N, then the pairs formed from the smallest and largest elements are also less than N.

The paper generalizes bounding conditions to both upper and lower bounds in totally ordered semi-groups.

Provides a framework for solving bounding problems in algebraic structures with order and operation constraints.

Abstract

The following is shown : Let $S=\{a_1,a_2,..,a_{2n}\}$ be a subset of a totally ordered commutative semi-group $(G,*,\leq)$ with $a_1\leq a_2\leq...\leq a_{2n}$. Provided that a system of $n$ $a_{i_k} * a_{j_k}\ (a_{i_k}, a_{j_k} \in G ;\ 1 \leq k \leq n)$, where all $2n$ elements in $S$ must be used, are less than an element $N\ (\in G)$, then $a_1*a_{2n}, a_2*a_{2n-1},..., a_n*a_{n+1}$ are all less than $N$. This may be called the Upper Bounding Case. Moreover in the same way, we shall treat also the Lower Bounding Case.