Cylindric quasi-implication algebras

TL;DR

This paper introduces cylindric quasi-implication algebras, establishing their equivalence with quantum cylindric algebras, and provides new constructions of cylindric orthoframes from these algebras, bridging algebraic and quantum logic.

Contribution

It demonstrates the categorical equivalence between cylindric quasi-implication algebras and quantum cylindric algebras, and introduces two novel methods to construct cylindric orthoframes from these algebras.

Findings

Equivalence between cylindric quasi-implication algebras and quantum cylindric algebras.

Two new constructions of cylindric orthoframes from algebraic structures.

Abstract

In this note, we study the operation of Sasaki hook within the setting of quantum cylindric algebras by introducing cylindric quasi-implication algebras. It is first demonstrated that every quantum cylindric algebra can be converted into a cylindric quasi-implication algebra and conversely that every cylindric quasi-implication algebra gives rise to a quantum cylindric algebra. These constructions are then shown to induce an isomorphism between the category $\mathbf{CQIA}$ of cylindric quasi-implication algebras and the category $\mathbf{QCA}$ of quantum cylindric algebras. We then give two alternative constructions of a cylindric orthoframe $X_A$ from a cylindric quasi-implication algebra $A$. The first construction of $X_A$ arises via the non-zero elements of $A$ and generalizes the construction given by Harding in the setting of cylindric ortholattices from the perspective of MacNeille completions. The second construction of $X_A$ arises via the proper filters of $A$ and generalizes the construction given by McDonald in the setting of cylindric ortholattices from the perspective of canonical completions.