On Bounding Problems on Totally Ordered Commutative Semi-Groups

The following is shown : Let $S=\{a_1,a_2,\mathrm{.}\mathrm{.},a_{2n}\}$ be a subset of a totally ordered commutative semi-group $(G,\ast ,\le )$ with $a_1\leq a_2\leq…\leq a_{2n}$\mathrm{.}Providedthatasystemof$n$ $a_{i_k} * a_{j_k}
(a_{i_k}, a_{j_k} \in G ;\ 1 \leq k \leq n)$,whereall$2n$elementsin$S$mustbeused,arelessthananelement$N\ (\in G)$,then$a_1a_{2n}, a_2a_{2n-1},…, a_n*a_{n+1}$arealllessthan$N$. This may be called the Upper Bounding Case. Moreover in the same way, we shall treat also the Lower Bounding Case.