On the Riesz structures of a lattice ordered abelian group

2019 ◽  
Vol 69 (6) ◽  
pp. 1237-1244
Author(s):  
Giacomo Lenzi
Keyword(s):  
Abelian Group ◽  
Model Theory ◽  
Positive Element ◽  
Space Structure ◽  
Real Vector ◽  
Ordered Group ◽  
Positive Real ◽  
Strong Unit ◽  
Totally Ordered Group ◽  
Mv Algebras

Abstract A Riesz structure on a lattice ordered abelian group G is a real vector space structure where the product of a positive element of G and a positive real is positive. In this paper we show that for every cardinal k there is a totally ordered abelian group with at least k Riesz structures, all of them isomorphic. Moreover two Riesz structures on the same totally ordered group are partially isomorphic in the sense of model theory. Further, as a main result, we build two nonisomorphic Riesz structures on the same l-group with strong unit. This gives a solution to a problem posed by Conrad in 1975. Finally we apply the main result to MV-algebras and Riesz MV-algebras.

Weakly atomic-compact relational structures

1971 ◽  
Vol 36 (1) ◽  
pp. 129-140 ◽  
Author(s):  
G. Fuhrken ◽  
W. Taylor
Keyword(s):  
Abelian Group ◽  
Model Theory ◽  
Compact Abelian Group ◽  
Finite Subset ◽  
Relational Structure ◽  
Similarity Type ◽  
First Order ◽  
Relational Structures ◽  
Order Language ◽  
Weakly Atomic

A relational structure is called weakly atomic-compact if and only if every set Σ of atomic formulas (taken from the first-order language of the similarity type of augmented by a possibly uncountable set of additional variables as “unknowns”) is satisfiable in whenever every finite subset of Σ is so satisfiable. This notion (as well as some related ones which will be mentioned in §4) was introduced by J. Mycielski as a generalization to model theory of I. Kaplansky's notion of an algebraically compact Abelian group (cf. [5], [7], [1], [8]).

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.

scholarly journals Representation of perfect and local MV-algebras

2011 ◽  
Vol 61 (3) ◽  
Author(s):  
Brunella Gerla ◽  
Ciro Russo ◽  
Luca Spada
Keyword(s):  
Unit Interval ◽  
Representation Theorems ◽  
Strong Unit ◽  
Mv Algebras

AbstractWe describe representation theorems for local and perfect MV-algebras in terms of ultraproducts involving the unit interval [0, 1]. Furthermore, we give a representation of local Abelian ℓ-groups with strong unit as quasi-constant functions on an ultraproduct of the reals. All the above theorems are proved to have a uniform version, depending only on the cardinality of the algebra to be embedded, as well as a definable construction in ZFC.The paper contains both known and new results and provides a complete overview of representation theorems for such classes.

scholarly journals Pseudo MV-algebras are intervals in ℓ-groups

2002 ◽  
Vol 72 (3) ◽  
pp. 427-446 ◽  
Author(s):  
Anatolij Dvurečenskij
Keyword(s):  
Strong Unit ◽  
Famous Result ◽  
Mv Algebra ◽  
Mv Algebras

AbstractWe show that any pseudo MV-algebra is isomorphic with an interval Γ(G, u), where G is an ℓ-group not necessarily Abelian with a strong unit u. In addition, we prove that the category of unital ℓ-groups is categorically equivalent with the category of pseudo MV-algebras. Since pseudo MV-algebras are a non-commutative generalization of MV-algebras, our assertions generalize a famous result of Mundici for a representation of MV-algebras by Abelian unital ℓ-groups. Our methods are completely different from those of Mundici. In addition, we show that any Archimedean pseudo MV-algebra is an MV-algebra.

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 products of topological groups

1987 ◽  
Vol 102 (2) ◽  
pp. 281-295
Author(s):  
M. Henriksen ◽  
R. Kopperman ◽  
F. A. Smith
Keyword(s):  
Rational Numbers ◽  
Lexicographic Order ◽  
Total Order ◽  
Topological Groups ◽  
Product Topology ◽  
Ordered Group ◽  
Interval Topology ◽  
Ordered Groups ◽  
Totally Ordered Group ◽  
Usual Order

The topology most often used on a totally ordered group (G, <) is the interval topology. There are usually many ways to totally order G x G (e.g., the lexicographic order) but the interval topology induced by such a total order is rarely used since the product topology has obvious advantages. Let ℝ(+) denote the real line with its usual order and Q(+) the subgroup of rational numbers. There is an order on Q x Q whose associated interval topology is the product topology, but no such order on ℝ x ℝ can be found. In this paper we characterize those pairs G, H of totally ordered groups such that there is a total order on G x H for which the interval topology is the product topology.

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