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Duality Theorems for Finitely Generated Vector Lattices
Author(s) -
Bey W. M.
Publication year - 1975
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/s3-31.1.114
Subject(s) - singleton , mathematics , duality (order theory) , citation , combinatorics , library science , computer science , pregnancy , genetics , biology
In the third section of the paper, by applying a theorem proved by the author in (2), we show that 'Z-map' and 'piecewise homogeneous linear function' are equivalent concepts within the subcategory of V whose objects are euclidean closed polyhedral cones. By exploiting this result, we then show that the polyhedral category is equivalent to the dual of the category whose objects are finitely generated projective vector lattices with a distinguished positive strong order unit e, together with such Z-morphisms between these vector lattices as preserve the distinguished element e. Although, throughout the paper, results are expressed in terms of vector lattices, there are, in many instances, generalizations to lattice-ordered abelian groups. In particular, most of the methods and results of the first two sections can easily be modified so as to apply in that context. Partial generalizations of the results of §§3 and 4 to lattice-ordered abelian groups will be the subject of a companion paper. 0. Preliminaries