Constructive Pointfree Topology Eliminates Non-constructive Representation Theorems from Riesz Space Theory

Abstract

In Riesz space theory it is good practice to avoid representation theorems which depend on the axiom of choice. Here we present a general methodology to do this using pointfree topology. To illustrate the technique we show that Archimedean almost f-algebras are commutative. The proof is obtained relatively straightforward from the proof by Buskes and van Rooij by using the pointfree Stone-Yosida representation theorem by Coquand and Spitters. Keywords Formal topology · Axiom of choice · Riesz space · Constructive analysisThe Stone-Yosida representation theorem [25,28] for Riesz spaces shows how to embed every Riesz space [20,29] into the Riesz space of continuous functions on its spectrum.Definition 1 A Riesz space is a vector space with compatible lattice operationsi.e. f ∧ g + f ∨ g = f + g and if f 0 and a 0, then af 0. A (strong) unit 1 in a Riesz space is an element such that for all f there exists a natural number n such that | f | n · 1. A Riesz space is Archimedean if for all n, n|x| y implies that x = 0.A Riesz space with a strong unit is Archimedean.Definition 2 A representation of a Riesz space R with unit is a linear map σ : R → Ê which respects the lattice operations and the unit. We denote by the space of representations equipped with the weakest topology which makes all the maps e r (σ ) := σ (r) continuous.