Characterisations of semi-prime Archimedean f-algebras (Q1100223)

It is well-known that an Archimedean Riesz space with a positive multiplication that is distributive over addition and such that \(x\perp y\Rightarrow (ax\perp y\) and xa\(\perp y)\) must be both associative and commutative. We show that if we assume that the algebra is associative and semi-prime then a one sided condition (x\(\perp y\Rightarrow ax\perp y\) or \(x\perp y\Rightarrow xa\perp y)\) forces commutativity, as does the condition \(x\perp y\Rightarrow abx\perp y\).