Mathematics Faculty Articles
Conrad Frames
Document Type
Article
Publication Date
9-1-2011
Publication Title
Topology and its Applications
Keywords
Frames and frame homomorphisms, Pairwise splitting l-groups and frames, Disjointification, o-Conrad frames
ISSN
0166-8641
Volume
158
Issue/No.
14
First Page
1875
Last Page
1887
Abstract
A Conrad frame is a frame which is isomorphic to the frame C(G) of all convex ℓ-subgroups of some lattice-ordered group G. It has long been known that Conrad frames have the disjointification property. In this paper a number of properties are considered that strengthen the disjointification property; they are referred to as the Conrad conditions. A particularly strong form of the disjointification property, the C-frame condition, is studied in detail. The class of lattice-ordered groups G for which C(G) is a C-frame is shown to coincide with the class of pairwise splitting ℓ-groups. The arguments are mostly frame-theoretic and Choice-free, until one tackles the question of whether C-frames are Conrad frames. They are, but the proof is decidedly not point-free. This proof actually does more: it shows that every algebraic frame with the FIP and disjointification can be coherently embedded in a C-frame. When the discussion is restricted to normal-valued lattice-ordered groups, one is able to produce examples of coherent frames having disjointification, which are not Conrad frames
NSUWorks Citation
Iberkleid, Wolf; Martinez, Jorge; and McGovern, Warren William, "Conrad Frames" (2011). Mathematics Faculty Articles. 45.
https://nsuworks.nova.edu/math_facarticles/45
DOI
10.1016/j.topol.2011.06.024
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Comments
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