Fully ordered groups in partially ordered linear algebras. (Q1771849)

In the classical theory of ordered groups a full (or totally) ordered group is a group \(G\) endowed with a total order \(\leq \) satisfying the monotonicity law (ML), i.e.\ \(x\leq y\) implies \(xz\leq yz\) and \(zx\leq zy\) for all elements \(x,y,z\in G\). By Hölder's fundamental result, if \(G\) has the archimedean property then \(G\) is commutative. The author demonstrates that there are natural instances of groups with full ordering in which the ML fails. Especially he deals with (multiplicative) subgroups of any Dedekind \(\sigma \)-complete partially ordered real associative linear algebra (dsc-pola) \(A\) which are as sets fully ordered with respect to the order of \(A\). Such subgroups are called chained. The main result of the paper is: A chained group \(G\) in a dsc-pola \(A\) is commutative. Further, general properties of such groups are discussed.