Singular Archimedean lattice-ordered groups

1998 ◽  
Vol 40 (2) ◽  
pp. 119-147 ◽  
Author(s):  
A. W. Hager ◽  
J. Martinez
Keyword(s):  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Archimedean Lattice

A theorem and a question about epicomplete archimedean lattice-ordered groups

2009 ◽  
Vol 62 (2-3) ◽  
pp. 165-184 ◽  
Author(s):  
R. N. Ball ◽  
A. W. Hager ◽  
D. G. Johnson ◽  
A. Kizanis
Keyword(s):  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Archimedean Lattice

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.

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.

Hölder categories

2014 ◽  
Vol 64 (3) ◽  
Author(s):  
Anthony Hager ◽  
Jorge Martínez
Keyword(s):  
Order Unit ◽  
Real Numbers ◽  
Initial Element ◽  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Terminal Object ◽  
Epireflective Hull ◽  
Strong Order ◽  
Archimedean Lattice ◽  
Initial Object

AbstractHölder categories are invented to provide an axiomatic foundation for the study of categories of archimedean lattice-ordered algebraic structures. The basis of such a study is Hölder’s Theorem (1908), stating that the archimedean totally ordered groups are precisely the subgroups of the additive real numbers ℝ with the usual addition and ordering, which remains the single most consequential result in the studies of lattice-ordered algebraic systems since Birkhoff and Fuchs to the present.This study originated with interest in W*, the category of all archimedean lattice-ordered groups with a designated strong order unit, and the ℓ-homomorphisms which preserve those units, and, more precisely, with interest in the epireflections on W*. In the course of this study, certain abstract notions jumped to the forefront. Two of these, in particular, seem to have been mostly overlooked; some notion of simplicity appears to be essential to any kind of categorical study of W*, as are the quasi-initial objects in a category. Once these two notions have been brought into the conversation, a Hölder category may then be defined as one which is complete, well powered, and in which(a) the initial object I is simple, and(b) there is a simple quasi-initial coseparator R.In this framework it is shown that the epireflective hull of R is the least monoreflective class. And, when I = R — that is, the initial element is simple and a coseparator — a theorem of Bezhanishvili, Morandi, and Olberding, for bounded archimedean f-algebras with identity, can be be generalized, as follows: for any Hölder category subject to the stipulation that the initial object is a simple coseparator, every uniformly nontrivial reflection — meaning that the reflection of each non-terminal object is non-terminal — is a monoreflection.Also shown here is the fact that the atoms in the class of epireflective classes are the epireflective hulls of the simple quasi-initial objects. From this observation one easily deduces a converse to the result of Bezhanishvili, Morandi, and Olberding: if in a Hölder category every epireflection is a monoreflection, then the initial object is a coseparator.

Comments on monoreflections in unital archimedean lattice-ordered groups and rings

2016 ◽  
Vol 39 (8) ◽  
pp. 1093-1099 ◽  
Author(s):  
Anthony W. Hager ◽  
James J. Madden
Keyword(s):  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Archimedean Lattice
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