Square closed pointed vector lattices

Abstract Given an Archimedean vector lattice E , we present one elementary property of E which is equivalent to the entire traditional list of axioms which makes E a $$\Phi $$ Φ -algebra. We call a vector lattice with this property “square closed”. More generally, we then introduce the notion of a pseudo square closed vector lattice and prove that an Archimedean vector lattice is a semiprime f -algebra if and only if it is pseudo square closed. Examples are also provided to show that the assumption of “Archimedean” cannot be dropped. This theory serves as an efficient tool for determining whether or not an Archimedean vector lattice is a $$\Phi $$ Φ -algebra (or a semiprime f -algebra). To illustrate this point, we generalize a well-known result for uniformly complete Archimedean vector lattices with a strong order unit by proving that every functionally complete Archimedean vector lattice with a strong order unit is a $$\Phi $$ Φ -algebra.