Commutativity of some Archimedean ordered algebras

Kouki, Naoual; Toumi, Mohamed Ali; Toumi, Nedra

doi:10.1007/s11117-014-0277-5

Positivity

2014

DOI: 10.1007/s11117-014-0277-5

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Commutativity of some Archimedean ordered algebras

Naoual Kouki

Mohamed Ali Toumi

Nedra Toumi

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2020

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Equivalence of order and algebraic properties in ordered $$^*$$-algebras

Schötz

2020

Positivity

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Consider an ordered abelian group G (i.e. an abelian group with a translation invariant partial order) endowed with a biadditive operation µ : G×G → G. We will show that under some additional assumptions concerning the compatibility between the operation µ and the order, the operation µ is automatically both associative and commutative, and all squares µ(g, g) with g ∈ G are positive. These additional assumptions essentially are that the order is (integrally) closed (sometimes also referred to as "Archimedean", especially in the context of lattice-ordered groups) and localizable (the structure of the set of positive elements of G does not obstruct adjoining multiplicative inverses of "strictly" positive elements). Some results of this type have already been obtained by various authors -some of them more than half a century ago -in much more restricitve contexts, especially in the uniformly bounded and the lattice-ordered cases; we discuss how most of these classical results can be obtained from our main theorem. We also prove that a partially ordered skew field, in which there is no element that dominates all natural numbers, is a field.

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Equivalence of order and algebraic properties in ordered $$^*$$-algebras

Schötz

2020

Positivity

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Commutativity of some Archimedean ordered algebras

Cited by 1 publication

References 9 publications

Equivalence of order and algebraic properties in ordered $$^*$$-algebras

Equivalence of order and algebraic properties in ordered $$^*$$-algebras

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