Fully ordered groups in partially ordered linear algebras

Dai, Taen-Yu

doi:10.1007/s000120050222

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Introduction and Basic Definitions1

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2015

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“…We will assume throughout that all partially ordered linear algebras are Dedekind σ-complete, and refer to such an algebra simply as a POLA. Alternative names in the literature include polac [2] or dsc-pola [3]. A natural example of a POLA is the algebra of real square matrices of some fixed order n with partial order defined entrywise.…”

Section: Introduction and Basic Definitionsmentioning

confidence: 99%

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

Dai

2015

Algebra Univers.

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

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Section: Introduction and Basic Definitionsmentioning

confidence: 99%