A study of a special class of groups in a partially ordered linear algebra

Abstract

We study subgroups G of the multiplicative group of a unital Dedekind ${\sigma}$-complete partially ordered linear algebra A satisfying the property that the set ${H = \{ x^{2} : x \in G \}}$ is a chain. Among other things, we show that if such a group is bounded above by some element ${u \in A}$, then x ⁴ = 1 for all ${x \in G}$ (so that H consists entirely of involutions), while if the upper bound u belongs to G, then x ² = 1 for all ${x \in G}$ (so that G is abelian).