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.

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°).

Non-amalgamation of ordered groups

1984 ◽  
Vol 95 (2) ◽  
pp. 191-195 ◽  
Author(s):  
A. M. W. Glass ◽  
D. Saracino ◽  
C. Wood
Keyword(s):  
Positive Integer ◽  
Total Order ◽  
Ordered Group ◽  
Orderable Group ◽  
Ordered Groups

An ordered group (o-group for short) is a group endowed with a linear (i.e. total) order such that for all x, y, z, xz ≤ yz and zx ≤ zy whenever x ≤ y. A group for which such an order exists is called an orderable group. A group G is said to be divisible if for each positive integer m and each g ε G, there is x ε G such that xm = g.

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].

An Application of Ultraproducts to Lattice-Ordered Groups

1972 ◽  
Vol 24 (6) ◽  
pp. 1063-1064
Author(s):  
A. M. W. Glass
Keyword(s):  
Total Order ◽  
Lattice Ordered Group ◽  
Independent Subset ◽  
Ordered Group ◽  
Ordered Groups ◽  
Partial Converse ◽  
Lattice Ordered Groups ◽  
Total Orders

Using ultraproducts, N. R. Reilly proved that if G is a representable lattice-ordered group and J is an independent subset totally ordered by ≺, then the order on G can be extended to a total order which induces ≺ on J (see [5]). In [4], H. A. Hollister proved that a group G admits a total order if and only if it admits a representable order and, moreover, every latticeorder on a group is the intersection of right total orders. The purpose of this paper is to provide a partial converse, viz: if G is a lattice-ordered group and J is an independent subset totally ordered by ≺, then the order on G can be extended to a right total order which induces ≺ on J.

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.

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.

scholarly journals Convergence in partially ordered groups

1973 ◽  
Vol 18 (3) ◽  
pp. 239-246
Author(s):  
Andrew Wirth
Keyword(s):  
Uniform Convergence ◽  
Banach Lattice ◽  
Order Convergence ◽  
Ordered Group ◽  
Partially Ordered ◽  
Partially Ordered Group ◽  
Ordered Groups ◽  
Integrally Closed ◽  
Relative Uniform Convergence ◽  
New Type

AbstractRelative uniform limits need not be unique in a non-archimedean partially ordered group, and order convergence need not imply metric convergence in a Banach lattice. We define a new type of convergence on partially ordered groups (R-convergence), which implies both the previous ones, and does not have these defects. Further R-convergence is equivalent to relative uniform convergence on divisible directed integrally closed partially ordered groups, and to order convergence on fully ordered groups.

On Integration in Partially Ordered Groups

1983 ◽  
Vol 35 (2) ◽  
pp. 353-372 ◽  
Author(s):  
Panaiotis K. Pavlakos
Keyword(s):  
Vector Spaces ◽  
Order Structure ◽  
Topological Groups ◽  
Topological Vector Spaces ◽  
Lattice Group ◽  
Partially Ordered ◽  
Ordered Groups ◽  
Measurable Functions ◽  
Ordered Vector Spaces ◽  
Definition Of

M. Sion and T. Traynor investigated ([15]-[17]), measures and integrals having values in topological groups or semigroups. Their definition of integrability was a modification of Phillips-Rickart bilinear vector integrals, in locally convex topological vector spaces.The purpose of this paper is to develop a good notion of an integration process in partially ordered groups, based on their order structure. The results obtained generalize some of the results of J. D. M. Wright ([19]-[22]) where the measurable functions are real-valued and the measures take values in partially ordered vector spaces.Let if be a σ-algebra of subsets of T, X a lattice group, Y, Z partially ordered groups and m : H → F a F-valued measure on H. By F(T, X), M(T, X), E(T, X) and S(T, X) are denoted the lattice group of functions with domain T and with range X, the lattice group of (H, m)-measurable functions of F(T, X) and the lattice group of (H, m)-elementary measurable functions of F(T, X) and the lattice group of (H, m)-simple measurable functions of F(T, X) respectively.

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.

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