Square closed pointed vector lattices

TL;DR

This paper introduces the concept of square closed vector lattices, providing a new criterion to identify $ ext{Phi}$-algebras and semiprime $f$-algebras within Archimedean vector lattices, simplifying their classification.

Contribution

It defines square closed and pseudo square closed vector lattices, establishing their equivalence to $ ext{Phi}$-algebras and semiprime $f$-algebras, respectively, and generalizes a key result on functionally complete lattices.

Findings

Square closed property characterizes $ ext{Phi}$-algebras.

Pseudo square closed property characterizes semiprime $f$-algebras.

Every functionally complete Archimedean vector lattice with a strong order unit is a $ ext{Phi}$-algebra.

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 $\Phi$-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. This theory serves as an efficient tool for determining whether or not an Archimedean vector lattice is a $\Phi$-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 $\Phi$-algebra.