On Bounding Problems on Totally Ordered Commutative Semi-Groups

Abstract

The following is shown : Let $S=\{a_1,a_2,..,a_{2n}\}$ be a subset of a totally ordered commutative semi-group $(G,\ast ,\le )$ with $a_1\le a_2\le ...\le a_{2n}$. Provided that a system of $n$ $a_{i_k}\ast a_{j_k}(a_{i_k},a_{j_k}\in G;1\le k\le n)$, where all $2n$ elements in $S$ must be used, are less than an element $N(\in G)$, then $a_1\ast a_{2n},a_2\ast a_{2n-1},...,a_n\ast a_{n+1}$ 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.

10 pages