Lateral completion and structure sheaf of an archimedean l-group (original) (raw)

Hulls for Various Kinds of α-Completeness in Archimedean Lattice-Ordered Groups

A Journal on The Theory of Ordered Sets and Its Applications, 1999

Within Archimedean l-groups, and with a an infinite cardinal or 8, we consider X-hulls where X stands for any of the following classes of l-groups: a-projectable; laterally a-complete; boundedly laterally a-complete; conditionally a-complete; combinations of the preceding, together with divisibility and/or relative uniform completeness. All these hulls exist, and may be obtained by iterated adjunction of the required extra elements,

THE STRUCTURE OF THE alpha-COMPLETION OF A LATTICE ORDERED GROUP

1989

We prove that, roughly speaking, the •-completion G i " is the least extension of a lattice ordered group (&group) G in which G is large and which is order closed in every extension i which it (G ia) is large. Numerous related completion results are proved: in addition, we obtain fairly detailed structural descriptions ofG i • in the cases when G is completely distributive, archimedean or strongly projectable. The study of •-congergence was begun by Papangelou [27], carried forward in Ellis ' thesis [22], and culminated in a paper of Madell [25]. Ball and Davis [10] used the general Cauchy completion techniques of [2] to prove the existence and uniqueness of the c•'-completion G i". In addition, many of these ideas have been developed in the broader context of distribu-tive lattices in [6]. This paper, whose purpose is to explicate the structure of G i•, is motivated by the opinion that •-completeness is a natural and important lattice property with interesting c...

On the Aspects of Enriched Lattice-valued Topological Groups and Closure of Lattice-valued Subgroups

European Journal of Pure and Applied Mathematics

Starting with L as an enriched cl-premonoid, in this paper, we explore some categorical connections between L-valued topological groups and Kent convergence groups, where it is shown that every L-valued topological group determines a well-known Kent convergence group, and conversely, every Kent convergence group induces an L-valued topological group. Considering an L-valued subgroup of a group, we show that the category of L-valued groups, L-GRP has initial structure. Furthermore, we consider a category L-CLS of L-valued closure spaces, obtaining its relation with L-valued Moore closure, and provide examples in relation to L-valued subgroups that produce Moore collection. Here we look at a category of L-valued closure groups, L-CLGRP proving that it is a topological category. Finally, we obtain a relationship between L-GRP and L-TransTOLGRP, the category of L-transitive tolerance groups besides adding some properties of L-valued closures of L-valued subgroups on L-valued topological...

Epicompletetion of archimedean l–groups and vector lattices with weak unit

Journal of the Australian Mathematical Society, 1990

In the category W of archimedean l–groups with distinguished weak order unit, with unitpreserving l–homorphism, let B be the class of W-objects of the form D(X), with X basically disconnected, or, what is the same thing (we show), the W-objects of the M/N, where M is a vector lattice of measurable functions and N is an abstract ideal of null functions. In earlier work, we have characterized the epimorphisms in W, and shown that an object G is epicomplete (that is, has no proper epic extension) if and only if G ∈ B. This describes the epicompletetions of a give G (that is, epicomplete objects epically containing G). First, we note that an epicompletion of G is just a “B-completion”, that is, a minimal extension of G by a B–object, that is, by a vector lattice of measurable functions modulo null functions. (C[0, 1] has 2c non-eqivalent such extensions.) Then (we show) the B–completions, or epicompletions, of G are exactly the quotients of the l–group B(Y(G)) of real-valued Baire funct...

Construction of existentially closed Abelian lattice-ordered groups using upper extensions

Algebra universalis, 2018

The upper extension construction of Ball, Conrad, and Darnel is used to produce new examples of non-Archimedean existentially closed Abelian lattice-ordered groups and boundedly existentially closed Abelian lattice-ordered groups. Also given are conditions under which an upper extension of a projectable Abelian lattice-ordered group is projectable.

The alpha\alpha alpha-completion of a lattice ordered group

Czechoslovak Mathematical Journal, 1983

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Epimorphic adjunction of a weak order unit to an Archimedean lattice-ordered group

Proceedings of the American Mathematical Society, 1992

It is shown that an archimedean l l -group G G can be embedded into another, H H , which has a weak unit, by an embedding that is epimorphic in archimedean l l -groups if and only if there is countable A ⊆ G A \subseteq G with A ⊥ = ( 0 ) {A^ \bot } = (0) . Then the extension H H can always be chosen conditionally and laterally σ \sigma -complete and the embedding essential, but can never be generated by G G together with finitely many extra elements unless G G already had a weak unit.