PBZ*-Lattices: Structure Theory and Subvarieties

TL;DR

This paper develops the structure theory of PBZ*-lattices, a class of algebraic structures that generalize orthomodular lattices and Kleene algebras, with applications in quantum mechanics foundations.

Contribution

It introduces foundational aspects of PBZ*-lattices, including ideals and central elements, and establishes key structure theorems and connections with other algebraic varieties.

Findings

Established basic theory of ideals and central elements in PBZ*-lattices

Proved structure theorems for PBZ*-lattices and subvarieties

Explored connections with subtractive and discriminator varieties

Abstract

We investigate the structure theory of the variety of \emph{PBZ*-lattices} and some of its proper subvarieties. These lattices with additional structure originate in the foundations of quantum mechanics and can be viewed as a common generalisation of orthomodular lattices and Kleene algebras expanded by an extra unary operation. We lay down the basics of the theories of ideals and of central elements in PBZ*-lattices, we prove some structure theorems, and we explore some connections with the theories of subtractive and binary discriminator varieties.