Abstract
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.
Highlights
The algebra of real-valued continuous functions C(X), for X a compact and Hausdorff space, has received great attention in all of its facets
In Di Nola and Leustean [18] it was noticed for the first time1 that the equivalence of Theorem 1.1 can be extended to an equivalence between the category of Riesz spaces with order unit and the category of certain equationally-definable algebras called Riesz MV–algebras (RMV–algebras, for short)
We prove that μ is a Riesz space homomorphism preserving the order unit
Summary
The algebra of real-valued continuous functions C(X), for X a compact and Hausdorff space, has received great attention in all of its facets (see, eg, Gillman and Jerison [25] and references therein). In Di Nola and Leustean [18] it was noticed for the first time that the equivalence of Theorem 1.1 can be extended to an equivalence between the category of Riesz spaces with order unit and the category of certain equationally-definable algebras called Riesz MV–algebras (RMV–algebras, for short). In this work we exploit this equivalence to propose a universal algebraic study of these structures This approach can be framed in a long list of successful attempts to use the tools of logic in functional analysis as for instance, the theory of approximate truth of.
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