Vector lattices in synaptic algebras

TL;DR

This paper investigates conditions under which a linear subspace of a synaptic algebra forms a vector lattice, establishing that pairwise commutativity is both necessary and sufficient when the subspace contains the identity and is closed under absolute value and carrier operations.

Contribution

It provides a characterization of vector lattices within synaptic algebras based on commutativity and closure properties, extending understanding of their structure.

Findings

V is a vector lattice iff its elements commute pairwise

Closure under absolute value and carrier is essential for lattice structure

Contains the identity element is a key condition

Abstract

A synaptic algebra $A$ is a generalization of the self-adjoint part of a von Neumann algebra. We study a linear subspace $V$ of $A$ in regard to the question of when $V$ is a vector lattice. Our main theorem states that if $V$ contains the identity element of $A$ and is closed under the formation of both the absolute value and the carrier of its elements, then $V$ is a vector lattice if and only if the elements of $V$ commute pairwise.