Stable \(f\)-algebras (Q1866830)

The paper under review deals with ideals (algebra ideals, regular ideals and order ideals) of Archimedean \(f\)-algebras. The author introduces the class of stable \(f\)-algebras and extends some results on Archimedean \(f\)-algebras with unit to Archimedean stable \(f\)-algebras. He gives necessary and sufficient conditions on Archimedean \(f\)-algebras to have the property that every algebra ideal is an order ideal and shows that every such \(f\)-algebra is stable. Analogous conditions are found for Archimedean \(f\)-algebras in which every regular algebra ideal is an order ideal. Furthermore, the paper contains representation theorems for stable \(f\)-algebras and \(f\)-algebras in which every algebra ideal is an order ideal.