Vector lattices in synaptic algebras

Abstract

AbstractA synaptic algebraAis a generalization of the self-adjoint part of a von Neumann algebra. We study a linear subspaceVofAin regard to the question of whenVis a vector lattice. Our main theorem states that ifVcontains the identity element ofAand is closed under the formation of both the absolute value and the carrier of its elements, thenVis a vector lattice if and only if the elements ofVcommute pairwise.