Open AccessSheaf representations and locality of Riesz spaces with order unit
Author(s) -
Antonio Di Nola,
Giacomo Lenzi,
Luca Spada
Publication year - 2021
Publication title -
journal of logic and analysis
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.278
H-Index - 4
ISSN - 1759-9008
DOI - 10.4115/jla.2021.13.2
Subject(s) - mathematics , sheaf , riesz representation theorem , pure mathematics , m. riesz extension theorem , riesz potential , hausdorff space , riesz transform , unit interval , algebra over a field , discrete mathematics
We present an algebraic study of Riesz spaces (=real vector lattices) with a (strong) order unit. We exploit a categorical equivalence between those structures and a variety of algebras called RMV-algebras. We prove two different sheaf representations for Riesz spaces with order unit: the first represents them as sheaves of linearly ordered Riesz spaces over a spectral space, the second represent them as sheaves of "local" Riesz spaces over a compact Hausdorff space. Motivated by the latter representation we study the class of local RMV-algebras. We study the algebraic properties of local RMV-algebra and provide a characterisation of them as special retracts of the real interval [0,1]. Finally, we prove that the category of local RMV-algebras is equivalent to the category of all Riesz spaces.