# Kadison's antilattice theorem for a synaptic algebra

## TL;DR

This paper proves that in a synaptic algebra with a complete orthomodular lattice of projections, the algebra is a factor if and only if it is an antilattice, extending Kadison's results on operator algebras.

## Contribution

It establishes a new equivalence between being a factor and an antilattice in synaptic algebras with complete projection lattices, generalizing Kadison's earlier work.

## Key findings

- A synaptic algebra is a factor iff it is an antilattice when the projection lattice is complete.
- Generalization of Kadison's results on infima and suprema in operator algebras.
- Provides new insights into the structure of synaptic algebras and their projection lattices.

## Abstract

We prove that if A is a synaptic algebra and the orthomodular lattice P of projections in A is complete, then A is a factor iff A is an antilattice. We also generalize several other results of R. Kadison pertaining to infima and suprema in operator algebras.