Cylindric quasi-implication algebras
Joseph McDonald
TL;DR
The paper addresses understanding the Sasaki hook in quantum cylindric algebras by introducing cylindric quasi-implication algebras, establishing bidirectional constructions between $\mathbf{QCA}$ and $\mathbf{CQIA}$, and proving an isomorphism between these categories. It develops two frame-construction approaches—MacLaren frames and Goldblatt frames—to obtain I-dimensional cylindric orthoframes from a cylindric quasi-implication algebra, generalizing MacNeille and canonical completions. The work demonstrates that the algebraic and relational representations align under the proposed translations and that the representations do not hinge on orthomodularity. Overall, it extends prior results on monadic and cylindric algebras to a cohesive framework for quantum cylindric logic with robust semantic duals.
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.