Spectral order on synaptic algebras

TL;DR

This paper introduces a spectral order on synaptic algebras, demonstrating that under certain conditions, these algebras form complete lattices and can be structured as Brouwer-Zadeh algebras, with De Morgan laws holding in finite types.

Contribution

It defines a new spectral order on synaptic algebras and explores its lattice properties, extending the algebraic structure and logical laws applicable to these algebras.

Findings

Spectral order makes A a Dedekind sigma-complete lattice.

Effect algebra E becomes a sigma-complete lattice under spectral order.

E can be structured as a Brouwer-Zadeh algebra in both orders.

Abstract

We define and study an alternative partial order, called the spectral order, on a synaptic algebra-a generalization of the self-adjoint part of a von Neumann algebra. We prove that if the synaptic algebra A is norm complete (a Banach synaptic algebra), then under the spectral order, A is Dedekind sigma-complete lattice, and the corresponding effect algebra E is a sigma-complete lattice. Moreover, E can be organized into a Brouwer-Zadeh algebra in both the usual (synaptic) and spectral ordering; and if A is Banach, then E is a Brouwer-Zadeh lattice in the spectral ordering. If A is of finite type, then De Morgan laws hold on E in both the synaptic and spectral ordering.