The Role of Frames in the Development of Lattice-ordered Groups: A Personal Account
A frame is a complete lattice in which finite meets distribute over arbitrary joins.
Frames have only recently made a formal entry into the development of lattice-ordered groups. On the other hand, the work of Paul Conrad and some of his students of the sixties and seventies, analyzing a lattice-ordered group through its lattice of convex ℓ-subgroups, is frame theory in disguise. In more recent work, pure frame theory has found application to problems in ℓ-groups, producing, in several cases, theorems which had not been possible with more traditional techniques. And now this turning of the tables has been taken a step further: proving theorems from the theory of ℓ-groups in frame-theoretic settings, without invoking the Axiom of Choice or other axioms which imply the existence of points in spectra.
This article aims to inform and convince the reader: inform, in broad terms, and convince that the phenomena discussed in the preceding paragraph constitute an honorable research activity. This is a survey article of modest length: selectivity is a must - with the choices of illustrations being left, for good or ill, to the taste and prejudices of the author.
The exposition is in three parts, following the three (chronological) aspects of the role of frame theory in the development of ℓ-groups. First up is the famous theorem of Conrad on finite-valued ℓ-groups. This is followed by an account of dimension theory, particularly as it applies to the ℓ-dimension of rings of continuous functions. Finally, there is an account of the recent and ongoing work on the epicompletion in a category of regular frames, and related issues concerning archimedean frames.
KeywordsRiesz Space Compact Element Frame Theory Essential Extension Algebraic Lattice
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