Lattice-ordered fields as convolution algebras

Redfield, R. H.

doi:10.1016/0021-8693(92)90158-i

Cited by 11 publications

(25 citation statements)

References 16 publications

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“…For any PROOF. This result follows from [19,Proposition 7.2] One obvious question raised by this result is when the algebra of condition (i) is a field. Using the results of Section 3, we have the following partial answer to this question.…”

Section: Embeddings Of Lattice-ordered Fieldsmentioning

confidence: 78%

“…Form the product n A T; for a € A and r e L, let a,r e]l A t be the vectors , {r a p), +, •), is a commutative ring with 7 = T as the multiplicative identity (cf. [4], [5], [13], [16], [20], [21], [22] [9], [13], [16], [17], [18], [19], [20], [21], [22]. When Ribenboim's convolution rings [20], [21], [22] are based on a group, they are a special case of this construction; he requires the supports to be locally inversely well-ordered and to have no infinite disjoint subsets.)…”

Section: Power Series Algebrasmentioning

confidence: 99%

“…There are several different proofs of this-see [14] (cf. [15]), [24], and [19]. In [19], the embeddability of L in a power series algebra over M(L) was characterized in over twenty different ways.…”

Section: Embeddings Of Lattice-ordered Fieldsmentioning

confidence: 99%

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Lattice-ordered power series fields

Redfield

1992

J Aust Math Soc A

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A lattice-ordered power series algebra of a totally ordered field over a rooted abelian group may be constructed in a way that is arbitrary only in requiring that a factor set be chosen in the field and an extended total order be chosen on the group modulo its torsion subgroup. The resulting algebra is a field if and only if the subalgebra of elements with torsion support form a field. It follows that if the torsion subgroup may be independently embedded in the algebraic closure of the totally ordered field, or if the resulting algebra has no zero-divisors, then the algebra is a field. The set of supporting subsets for the power series may be characterized abstractly in such a way that previous representation theorems of lattice-ordered fields into power series algebras may be applied to produce representations into power series fields.1991 Mathematics subject classification (Amer. Math. Soc): primary 06 F 25; secondary 06 F 15, 12 J 15, 13 J 05, 16 A 27, 16 A 86.

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Section: Embeddings Of Lattice-ordered Fieldsmentioning

confidence: 78%

Section: Power Series Algebrasmentioning

confidence: 99%

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Lattice-ordered power series fields

Redfield

1992

J Aust Math Soc A

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“…It is well known ( [2], [4], [5], [6]) that X T F is a field with respect to coordinatewise addition and to convolution as multiplication:…”

Section: Totally Ordered Subfields Of Partially Ordered Fieldsmentioning

confidence: 99%

“…It is well known that every -field L in which 1 > 0 has a maximal o-subfield M (L) (cf. [4], [7]). For archimedean -fields, Schwartz proved in [7] that even if 1 > 0, there is a subfield with properties similar to those of the o-subfields M (L).…”

Section: Introductionmentioning

confidence: 99%