Wreath Products of Nonoverlapping Lattice Ordered Groups

1975 ◽  
Vol 17 (5) ◽  
pp. 713-722 ◽  
Author(s):  
John A. Read
Keyword(s):  
Wreath Product ◽  
Wreath Products ◽  
General Question ◽  
Real Numbers ◽  
Ordered Group ◽  
The Real ◽  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Totally Ordered Group ◽  
Product Of Subgroups

One of the fundamental tools in the theory of totally ordered groups is Hahn’s Theorem (a detailed discussion may be found in Fuchs [3]), which asserts, roughly, that every abelian totally ordered group can be embedded in a lexicographically ordered (unrestricted) direct sum of copies of the ordered group of real numbers. Almost any general question regarding the structure of abelian totally ordered groups can be answered by reference to Hahn’s theorem. For the class of nonabelian totally ordered groups, a theorem which parallels Hahn’s Theorem is given in [5], and states that each totally ordered group can be o-embedded in an ordered wreath product of subgroups of the real numbers. In order to extend this theorem to include an “if and only if” statement, one must consider lattice ordered groups, as an ordered wreath product of subgroups of the real numbers is, in general, not totally-ordered, but lattice ordered.

The Hahn representation theorem for ℓ-groups in ZFA

2000 ◽  
Vol 65 (2) ◽  
pp. 519-524
Author(s):  
D. Gluschankof
Keyword(s):  
Set Theory ◽  
Representation Theorem ◽  
Group Structure ◽  
Ordered Group ◽  
Representation Theorems ◽  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Totally Ordered Group ◽  
Basic Concepts ◽  
The Axiom Of Choice

In [7] the author discussed the relative force —in the set theory ZF— of some representation theorems for ℓ-groups (lattice-ordered groups). One of the theorems not discussed in that paper is the Hahn representation theorem for abelian ℓ-groups. This result, originally proved by Hahn (see [8]) for totally ordered groups and half a century later by Conrad, Harvey and Holland for the general case (see [4]), states that any abelian ℓ-group can be embedded in a Hahn product of copies of R (the real line with its natural totally-ordered group structure). Both proofs rely heavily on Zorn's Lemma which is equivalent to AC (the axiom of choice).The aim of this work is to point out the use of non-constructible axioms (i.e., AC and weaker forms of it) in the proofs. Working in the frame of ZFA, that is, the Zermelo-Fraenkel set theory where a non-empty set of atoms is allowed, we present alternative proofs which, in the totally ordered case, do not require the use of AC. For basic concepts and notation on ℓ-groups the reader can refer to [1] and [2]. For set theory, to [11].

scholarly journals Minimal vector lattice covers

1971 ◽  
Vol 5 (3) ◽  
pp. 331-335 ◽  
Author(s):  
Roger D. Bleier
Keyword(s):  
Vector Lattice ◽  
Lattice Ordered Group ◽  
Ordered Group ◽  
Vector Lattices ◽  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Archimedean Lattice

We show that each archimedean lattice-ordered group is contained in a unique (up to isomorphism) minimal archimedean vector lattice. This improves a result of Paul F. Conrad appearing previously in this Bulletin. Moreover, we show that this relationship between archimedean lattice-ordered groups and archimedean vector lattices is functorial.

A Model of the Real Numbers

1963 ◽  
Vol 6 (2) ◽  
pp. 239-255
Author(s):  
Stanton M. Trott
Keyword(s):  
Irrational Number ◽  
Additive Group ◽  
Linear Ordering ◽  
Real Numbers ◽  
Ordered Group ◽  
The Real ◽  
Linear Orderings ◽  
Positive Elements ◽  
The Law ◽  
Ordered Pairs

The model of the real numbers described below was suggested by the fact that each irrational number ρ determines a linear ordering of J2, the additive group of ordered pairs of integers. To obtain the ordering, we define (m, n) ≤ (m', n') to mean that (m'- m)ρ ≤ n' - n. This order is invariant with group translations, and hence is called a "group linear ordering". It is completely determined by the set of its "positive" elements, in this case, by the set of integer pairs (m, n) such that (0, 0) ≤ (m, n), or, equivalently, mρ < n. The law of trichotomy for linear orderings dictates that only the zero of an ordered group can be both positive and negative.

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.

Archimedean Closures in Lattice-Ordered Groups

1969 ◽  
Vol 21 ◽  
pp. 1004-1012 ◽  
Author(s):  
Richard D. Byrd
Keyword(s):  
Partial Answer ◽  
Lattice Ordered Group ◽  
Ordered Group ◽  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Regular Subgroup

Conrad (10) and Wolfenstein (15; 16) have introduced the notion of an archimedean extension (a-extension) of a lattice-ordered group (l-group). In this note the class of l-groups that possess a plenary subset of regular subgroups which are normal in the convex l-subgroups that cover them are studied. It is shown in § 3 (Corollary 3.4) that the class is closed with respect to a-extensions and (Corollary 3.7) that each member of the class has an a-closure. This extends (6, p. 324, Corollary II; 10, Theorems 3.2 and 4.2; 15, Theorem 1) and gives a partial answer to (10, p. 159, Question 1). The key to proving both of these results is Theorem 3.3, which asserts that if a regular subgroup is normal in the convex l-subgroup that covers it, then this property is preserved by a-extensions.

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.

scholarly journals Tight Riesz groups

1972 ◽  
Vol 13 (2) ◽  
pp. 224-240 ◽  
Author(s):  
R. J. Loy ◽  
J. B. Miller
Keyword(s):  
General Theory ◽  
Open Interval ◽  
Discrete Topology ◽  
Topological Groups ◽  
Ordered Group ◽  
Partially Ordered ◽  
Interval Topology ◽  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Positive Elements

The theory of partially ordered topological groups has received little attention in the literature, despite the accessibility and importance in analysis of the group Rm. One obstacle in the way of a general theory seems to be, that a convenient association between the ordering and the topology suggests that the cone of all strictly positive elements be open, i.e. that the topology be at least as strong as the open-interval topology U; but if the ordering is a lattice ordering and not a full ordering then U itself is already discrete. So to obtain in this context something more interesting topologically than the discrete topology and orderwise than the full order, one must forego orderings which make lattice-ordered groups: in fact, the partially ordered group must be an antilattice, that is, must admit no nontrivial meets or joins (see § 2, 10°).

On a relative uniform completion of an archimedean lattice ordered group

2009 ◽  
Vol 59 (2) ◽  
Author(s):  
Štefan Černák ◽  
Judita Lihová
Keyword(s):  
Uniform Convergence ◽  
Subdirect Product ◽  
Lattice Ordered Group ◽  
Ordered Group ◽  
Vector Lattices ◽  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Archimedean Lattice ◽  
Finite Direct Product ◽  
Relatively Uniform Convergence

AbstractThe notion of a relatively uniform convergence (ru-convergence) has been used first in vector lattices and then in Archimedean lattice ordered groups.Let G be an Archimedean lattice ordered group. In the present paper, a relative uniform completion (ru-completion) $$ G_{\omega _1 } $$ of G is dealt with. It is known that $$ G_{\omega _1 } $$ exists and it is uniquely determined up to isomorphisms over G. The ru-completion of a finite direct product and of a completely subdirect product are established. We examine also whether certain properties of G remain valid in $$ G_{\omega _1 } $$. Finally, we are interested in the existence of a greatest convex l-subgroup of G, which is complete with respect to ru-convergence.

Relatively uniform convergences in archimedean lattice ordered groups

2010 ◽  
Vol 60 (4) ◽  
Author(s):  
Ján Jakubík ◽  
Štefan Černák
Keyword(s):  
Uniform Convergence ◽  
Vector Lattice ◽  
Radical Class ◽  
Ordered Group ◽  
Dedekind Completion ◽  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Divisible Hull ◽  
Archimedean Lattice ◽  
Relatively Uniform Convergence

AbstractFor an archimedean lattice ordered group G let G d and G∧ be the divisible hull or the Dedekind completion of G, respectively. Put G d∧ = X. Then X is a vector lattice. In the present paper we deal with the relations between the relatively uniform convergence on X and the relatively uniform convergence on G. We also consider the relations between the o-convergence and the relatively uniform convergence on G. For any nonempty class τ of lattice ordered groups we introduce the notion of τ-radical class; we apply this notion by investigating relative uniform convergences.

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.

Sign in / Sign up

Export Citation Format

Share Document