Quantum monadic algebras

TL;DR

This paper introduces quantum monadic and cylindric algebras, adapting classical algebraic logic structures to the quantum setting using von Neumann algebras, and explores their properties and relation to quantum predicate logic.

Contribution

It presents the first development of quantum analogs of monadic and cylindric algebras, extending algebraic logic into quantum theory.

Findings

Defined quantum monadic and cylindric algebras

Established their basic properties

Connected these algebras to quantum predicate logic

Abstract

We introduce quantum monadic and quantum cylindric algebras. These are adaptations to the quantum setting of the monadic algebras of Halmos, and cylindric algebras of Henkin, Monk and Tarski, that are used in algebraic treatments of classical and intuitionistic predicate logic. Primary examples in the quantum setting come from von Neumann algebras and subfactors. Here we develop the basic properties of these quantum monadic and cylindric algebras and relate them to quantum predicate logic.