Cylindric quasi-implication algebras

2025/11/21 by Joseph McDonald, McDonald, Joseph
Computer Science · Mathematics · #Advanced Algebra and Logic #Advanced Operator Algebra Research #Algebraic structures and combinatorial models #FOS: Mathematics #Logic (math.LO) #Rings and Algebras (math.RA)

paper · pdf · doi:10.48550/arxiv.2511.16968

openalex publication_date 2025/11/21 · openalex created_date 2025/11/25 · openalex updated_date 2026/07/28

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 CQIA of cylindric quasi-implication algebras and the category 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.

Citations

• Canonical completion and duality for cylindric ortholattices and cylindric Boolean algebras
• Orthogonality relations and operators on bounded quasi-implication algebras
• Functional monadic ortholattices and locally finite σ-free polyadic ortholattices
• Monadic ortholattices: completions and duality

Related

• Problems on neat embeddings solved by rainbow constructions and Monk algebras
• Orthogonality relations and operators on bounded quasi-implication algebras
• Canonical completion and duality for cylindric ortholattices and cylindric Boolean algebras
• A brief history of algebraic logic from neat embeddings to rainbow\n constructions
• Quasi-projective relation algebras and directed cylindric algebras are categorially equivalent