On the variety of Riesz spaces (Q2381835)

The class \({\mathcal R}\) of all Riesz spaces forms a variety of algebras in the universal algebra sense, hence it is closed under arbitrary products \((P)\), homomorphic images \((H)\) and subalgebras \((S)\). In one of the main results of the paper, the authors give a direct and elementary proof of the fact that \({\mathcal R}\) is generated as a variety by the Riesz space of the real numbers \(\mathbb{R}\). In fact, they prove the stronger result that \({\mathcal R}\) is generated as a quasivariety by \(\mathbb{R}\). A root system is a partially ordered set \(\Gamma\) for which for each \(\gamma \in \Gamma\), the set \(\{\delta \in \Gamma : \gamma \leq \delta \}\) is a chain. For a root system \(\Gamma\) in which all chains are finite, a Riesz space \(V (\Gamma, \mathbb{R})\) can be constructed. For a finitely generated Riesz space \(B\), \(\Gamma (B)\) is the set of proper prime ideals of \(B\) and \(V ( \Gamma (B), \mathbb{R})\) is the Riesz space based on the root system of proper prime ideals of \(B\). In the final result of the paper, the authors prove that finitely generated Riesz spaces are the subalgebras of real valued function spaces on root systems of finite height.