Square closed pointed vector lattices
Christopher Schwanke
TL;DR
This work reframes the axioms of $Φ$-algebras and semiprime $f$-algebras through an order-theoretic lens by introducing the square operation $f^{[2]}$ on pointed vector lattices with a unit $e$. It proves that an Archimedean vector lattice is a $Φ$-algebra if and only if the underlying pointed lattice $(E,e)$ is square closed, and that an Archimedean vector lattice is a semiprime $f$-algebra if and only if it is pseudo square closed. The authors define a canonical multiplication via $fg= frac12ig((f+g)^{[2]}-(f^{[2]}+g^{[2]})ig)$ and show that, in square-closed settings, this yields the usual $Φ$-algebra structure with unit $e$, linking order-theoretic square closure to algebraic properties. As a practical consequence, verifying square-closedness provides a streamlined tool for identifying $Φ$-algebras and semiprime $f$-algebras, and the paper extends known results by showing that functionally complete Archimedean vector lattices with a strong order unit are $Φ$-algebras. The framework also connects to a functional-calculus approach (via $h$-complete notions) to guarantee $Φ$-algebra structure in broader contexts.
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 $Φ$-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. This theory serves as an efficient tool for determining whether or not an Archimedean vector lattice is a $Φ$-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 $Φ$-algebra.