scholarly journals Divisibility of ordered groups

1972 ◽  
Vol 18 (1) ◽  
pp. 81-83
Author(s):  
I. W. Wright
Keyword(s):  
Abelian Group ◽  
Complete Lattice ◽  
Positive Cone ◽  
Real Numbers ◽  
Ordered Groups ◽  
Small Elements

In this paper it is shown that divisibility of a complete lattice ordered (abelian) group is closely related to the existence of a sufficient number of small elements in the positive cone.We shall denote the set of all real numbers by R which symbol will be reserved for this purpose. All terms used are as defined in Birkhoff(1). For the reader's convenience we now define the two terms most used in the sequel.

Existentially complete lattice-ordered groups

1980 ◽  
Vol 36 (3-4) ◽  
pp. 257-272 ◽  
Author(s):  
A. M. W. Glass ◽  
Keith R. Pierce
Keyword(s):  
Complete Lattice ◽  
Ordered Groups ◽  
Lattice Ordered Groups

Unbounded Positive Definite Functions

1969 ◽  
Vol 21 ◽  
pp. 1309-1318 ◽  
Author(s):  
James Stewart
Keyword(s):  
Abelian Group ◽  
Positive Measure ◽  
Positive Definite Function ◽  
Positive Definite ◽  
Locally Compact Abelian Group ◽  
Definite Function ◽  
Complex Numbers ◽  
Stieltjes Transform ◽  
Real Numbers ◽  
Image Position

Let G be an abelian group, written additively. A complexvalued function ƒ, defined on G, is said to be positive definite if the inequality1holds for every choice of complex numbers C1, …, cn and S1, …, sn in G. It follows directly from (1) that every positive definite function is bounded. Weil (9, p. 122) and Raïkov (5) proved that every continuous positive definite function on a locally compact abelian group is the Fourier-Stieltjes transform of a bounded positive measure, thus generalizing theorems of Herglotz (4) (G = Z, the integers) and Bochner (1) (G = R, the real numbers).If ƒ is a continuous function, then condition (1) is equivalent to the condition that2

Right-Ordered Polycyclic Groups

1974 ◽  
Vol 17 (2) ◽  
pp. 175-178 ◽  
Author(s):  
Roberta Botto Mura
Keyword(s):  
Additive Group ◽  
Real Numbers ◽  
Ordered Groups ◽  
Polycyclic Groups

One of the features that make right-ordered groups harder to investigate than ordered groups is that their system of convex subgroups may fail to have the following property:(*) if C and C’ are convex subgroups of G and C’ covers C, then C is normal in C’ and C’/C is order-isomorphic to a subgroup of the naturally ordered additive group of real numbers.

scholarly journals Real order-automorphism groups

1981 ◽  
Vol 31 (3) ◽  
pp. 257-261 ◽  
Author(s):  
E. C. Weinberg
Keyword(s):  
Automorphism Groups ◽  
Real Numbers ◽  
Order Automorphism ◽  
Ordered Groups ◽  
Lattice Ordered Groups

AbstractBy using the concept of tame embeddings of chains, a characterization is given of the subobjects of the lattice-ordered groups of order-automorphisms of the chains of rational and real numbers.

scholarly journals Cauchy completion of Abelian tight Riesz groups

1975 ◽  
Vol 19 (1) ◽  
pp. 62-73 ◽  
Author(s):  
B. F. Sherman
Keyword(s):  
Abelian Group ◽  
Vector Space ◽  
Important Case ◽  
Product Theorem ◽  
Real Vector ◽  
Partially Ordered ◽  
Word Group ◽  
Central Product ◽  
Ordered Groups ◽  
Cauchy Completion

This paper concerns the completions of partially ordered groups introduced by Fuchs (1965a) and the author (to appear); the p.o. groups under consideration are, generally, abelian tight Riesz groups, and so, throughout, the word “group” will refer to an abelian group.In section 3 we meet the cornerstone of the work, the central product theorem, by means of which we can interpret the Cauchy completion of a tight Riesz group in terms of the completion of any of its o-ideals; one particularly important case arises when the group has a minimal o-ideal. Such a minimal o-ideal is o-simple, and in section 6 the completion of an isolated o-simple tight Riesz group is shown to be a tight Riesz real vector space.

On o-minimal expansions of Archimedean ordered groups

1995 ◽  
Vol 60 (3) ◽  
pp. 817-831 ◽  
Author(s):  
Michael C. Laskowski ◽  
Charles Steinhorn
Keyword(s):  
Nonzero Element ◽  
Real Closed Field ◽  
Elementary Embedding ◽  
Closed Field ◽  
Real Numbers ◽  
Ordered Group ◽  
Ordered Groups ◽  
Differentiability Properties ◽  
Definable Function

AbstractWe study o-minimal expansions of Archimedean totally ordered groups. We first prove that any such expansion must be elementarily embeddable via a unique (provided some nonzero element is 0-definable) elementary embedding into a unique o-minimal expansion of the additive ordered group of real numbers . We then show that a definable function in an o-minimal expansion of enjoys good differentiability properties and use this to prove that an Archimedean real closed field is definable in any nonsemilinear expansion of . Combining these results, we obtain several restrictions on possible o-minimal expansions of arbitrary Archimedean ordered groups and in particular of the rational ordered group.

Un principe d'ax-kochen-ershov pour des structures intermédiates entre groupes et corps valués

1999 ◽  
Vol 64 (3) ◽  
pp. 991-1027 ◽  
Author(s):  
Françoise Delon ◽  
Patrick Simonetta
Keyword(s):  
Abelian Group ◽  
Cross Section ◽  
Additive Group ◽  
Ordered Group ◽  
Unary Predicate ◽  
Valued Fields ◽  
Valued Field ◽  
Ordered Groups ◽  
Existentially Closed ◽  
Image Position

AbstractAn Ax-Kochen-Ershov principle for intermediate structures between valued groups and valued fields.We will consider structures that we call valued B-groups and which are of the form 〈G, B, *, υ〉 where– G is an abelian group,– B is an ordered group,– υ is a valuation denned on G taking its values in B,– * is an action of B on G satisfying: ∀x ϵ G ∀ b ∈ B υ(x * b) = ν(x) · b.The analysis of Kaplanski for valued fields can be adapted to our context and allows us to formulate an Ax-Kochen-Ershov principle for valued B-groups: we axiomatise those which are in some sense existentially closed and also obtain many of their model-theoretical properties. Let us mention some applications:1. Assume that υ(x) = υ(nx) for every integer n ≠ 0 and x ϵ G, B is solvable and acts on G in such a way that, for the induced action, Z[B] ∖ {0} embeds in the automorphism group of G. Then 〈G, B, *, υ〉 is decidable if and only if B is decidable as an ordered group.2. Given a field k and an ordered group B, we consider the generalised power series field k((B)) endowed with its canonical valuation. We consider also the following structure:where k((B))+ is the additive group of k((B)), S is a unary predicate interpreting {Tb ∣ b ϵB}, and ×↾k((B))×S is the multiplication restricted to k((B)) × S, structure which is a reduct of the valued field k((B)) with its canonical cross section. Then our result implies that if B is solvable and decidable as an ordered group, then M is decidable.3. A valued B–group has a residual group and our Ax-Kochen-Ershov principle remains valid in the context of expansions of residual group and value group. In particular, by adding a residual order we obtain new examples of solvable ordered groups having a decidable theory.

scholarly journals Ordered groups with greatest common divisors theory

2000 ◽  
Vol 24 (7) ◽  
pp. 469-479
Author(s):  
Jiří Močkoř
Keyword(s):  
Abelian Group ◽  
Ordered Groups ◽  
Greatest Common Divisors

An embedding (called aGCD theory) of partly ordered abelian groupGinto abelian l-groupΓis investigated such that any element ofΓis an infimum of a subset (possible non-finite) fromG. It is proved that a GCD theory need not be unique. Acomplete GCD theoryis introduced and it is proved thatGadmits a complete GCD theory if and only if it admits a GCD theoryG→Γsuch thatΓis an Archimedean l-group. Finally, it is proved that a complete GCD theory is unique (up too-isomorphisms) and that a po-group admits the complete GCD theory if and only if anyv-ideal isv-invertible.

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