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.

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.

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.

scholarly journals C*-algebras generated by isometries and true representations

2019 ◽  
Vol 38 (5) ◽  
pp. 215-232
Author(s):  
Mamoon Ahmed
Keyword(s):  
The Other ◽  
Lattice Ordered Group ◽  
Covariant Representation ◽  
Ordered Group ◽  
Two Stage ◽  
C Algebras ◽  
Ordered Groups ◽  
Lattice Ordered Groups

Let (G; P) be a quasi-lattice ordered group. In this paper we present a modied proof of Laca and Raeburn's theorem about the covariant isometric representations of amenable quasi-lattice ordered groups [7, Theorem 3.7], by following a two stage strategy. First, we construct a universal covariant representation for a given quasi-lattice ordered group (G; P) and show that it is unique. The construction of this object is new; we have not followed either Nica's approach in [10] or Laca and Raeburn's approach in [7], although all three objects are essentially the same. Our approach is a very natural one and avoids some of the intricacies of the other approaches. Then we show if (G; P) is amenable, true representations of (G; P) generate C-algebras which are canonically isomorphic to the universal object.

Weak relatively uniform convergences on abelian lattice ordered groups

2011 ◽  
Vol 61 (5) ◽  
Author(s):  
Štefan Černák ◽  
Ján Jakubík
Keyword(s):  
Uniform Convergence ◽  
Lattice Ordered Group ◽  
Ordered Group ◽  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Abelian Lattice Ordered Group ◽  
Relative Uniform Convergence ◽  
Brouwerian Lattice ◽  
Abelian Lattice ◽  
Relatively Uniform Convergence

AbstractThe notion of relatively uniform convergence has been applied in the theory of vector lattices and in the theory of archimedean lattice ordered groups. Let G be an abelian lattice ordered group. In the present paper we introduce the notion of weak relatively uniform convergence (wru-convergence, for short) on G generated by a system M of regulators. If G is archimedean and M = G +, then this type of convergence coincides with the relative uniform convergence on G. The relation of wru-convergence to the o-convergence is examined. If G has the diagonal property, then the system of all convex ℓ-subgroups of G closed with respect to wru-limits is a complete Brouwerian lattice. The Cauchy completeness with respect to wru-convergence is dealt with. Further, there is established that the system of all wru-convergences on an abelian divisible lattice ordered group G is a complete Brouwerian lattice.

On a type of distributivity of lattice ordered groups

2014 ◽  
Vol 64 (2) ◽  
Author(s):  
Ján Jakubík
Keyword(s):  
Distributive Lattice ◽  
Boolean Algebras ◽  
Radical Class ◽  
Infinite Cardinal ◽  
Lattice Ordered Group ◽  
Ordered Group ◽  
Ordered Groups ◽  
Homogeneity Condition ◽  
Lattice Ordered Groups

AbstractLet m be an infinite cardinal. Inspired by a result of Sikorski on m-representability of Boolean algebras, we introduce the notion of r m-distributive lattice ordered group. We prove that the collection of all such lattice ordered groups is a radical class. Using the mentioned notion, we define and investigate a homogeneity condition for lattice ordered groups.

scholarly journals l-simple lattice-ordered groups

1974 ◽  
Vol 19 (2) ◽  
pp. 133-138 ◽  
Author(s):  
A. M. W. Glass
Keyword(s):  
Group Theory ◽  
Lattice Ordered Group ◽  
Normal Subgroups ◽  
Ordered Group ◽  
Central Task ◽  
Ordered Groups ◽  
Simple Lattice ◽  
Lattice Ordered Groups ◽  
Group Play

Let G be a lattice-ordered group (l-group) and H a subgroup of G. H is said to be an l-subgroup of G if it is a sublattice of G. H is said to be convex if h1, h2 ∈ H and h2 ≦ g ≦ h2 imply g ∈ H. The normal convex l-subgroups (l-ideals) of an l-group play the same role in the study of lattice-ordered groups as do normal subgroups in the investigation of groups. For this reason, an l-group is said to be l-simple if it has no non-trivial l-ideals. As in group theory, a central task in the examination of lattice-ordered groups is to characterise those l-groups which are l-simple.

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.

Countable lattice-ordered groups

1983 ◽  
Vol 94 (1) ◽  
pp. 29-33 ◽  
Author(s):  
A. M. W. Glass
Keyword(s):  
Free Lattice ◽  
Lattice Ordered Group ◽  
Ordered Group ◽  
Recursively Enumerable Set ◽  
Ordered Groups ◽  
Recursively Enumerable ◽  
Lattice Ordered Groups ◽  
Natural Analogues ◽  
Group Words ◽  
Combinatorial Group

A lattice-ordered group is a group and a lattice such that the group operation distributes through the lattice operations (i.e. f(g ∨ h)k = fgk ∨ fhk and dually). Lattice-ordered groups are torsion-free groups and distributive lattices. They further satisfy f ∧ g = (f−1 ∨ g−1)−1 and f ∨ g = (f−1 ∧ g−1)−1. Since the lattice is distributive, each lattice-ordered group word can be written in the form ∨A ∧B ωαβ where A and B are finite and each ωαβ is a group word in {xi: i ∈ I}. Unfortunately, even for free lattice-ordered groups, this form is not unique. We will use the prefix l- for maps between lattice-ordered groups that preserve both the group and lattice operations, and e for the identity element. A presentation (xi;rj(x) = e)i∈I, j∈J is the quotient of the free lattice-ordered group F on {xi: i∈I} by the l-ideal (convex normal sublattice subgroup) generated by its subset {rj(x): j ∈ J}. {xi: i ∈ I} is called a generating set and {ri(x):j∈J} a defining set of relations. If I and J are finite we have a finitely presented lattice-ordered group. If we can effectively enumerate all lattice-ordered group words r1(x), r2(x),… in xi; i∈I}. If I is finite and J (for this enumeration) is a recursively enumerable set, we say that we have a recursively presented lattice-ordered group. Throughout Z denotes the group of integers and ℝ the real line.Our purpose in this paper is to prove the natural analogues of three theorems from combinatorial group theory (5), chapter IV, theorems 4·9, 3·1 and 3·5-in particular, theorem C is a natural analogue of an unpublished theorem of Philip Hall (4).

Weak relatively uniform convergences on MV-algebras

2013 ◽  
Vol 63 (1) ◽  
Author(s):  
Štefan Černák ◽  
Ján Jakubík
Keyword(s):  
Present Article ◽  
Lattice Ordered Group ◽  
Ordered Group ◽  
Partially Ordered ◽  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Mv Algebra ◽  
Brouwerian Lattice ◽  
Mv Algebras ◽  
Natural Way

AbstractWeak relatively uniform convergences (wru-convergences, for short) in lattice ordered groups have been investigated in previous authors’ papers. In the present article, the analogous notion for MV-algebras is studied. The system s(A) of all wru-convergences on an MV-algebra A is considered; this system is partially ordered in a natural way. Assuming that the MV-algebra A is divisible, we prove that s(A) is a Brouwerian lattice and that there exists an isomorphism of s(A) into the system s(G) of all wru-convergences on the lattice ordered group G corresponding to the MV-algebra A. Under the assumption that the MV-algebra A is archimedean and divisible, we investigate atoms and dual atoms in the system s(A).

Lexico groups and direct products of lattice ordered groups

2016 ◽  
Vol 66 (1) ◽  
Author(s):  
Ján Jakubík
Keyword(s):  
Direct Products ◽  
Lattice Ordered Group ◽  
Ordered Group ◽  
Ordered Groups ◽  
Lattice Ordered Groups

AbstractA lattice ordered groupThen

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