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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2511.16968 |
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| _version_ | 1866918212624449536 |
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| author | McDonald, Joseph |
| author_facet | McDonald, Joseph |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_16968 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Cylindric quasi-implication algebras McDonald, Joseph Rings and Algebras Logic 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. |
| title | Cylindric quasi-implication algebras |
| topic | Rings and Algebras Logic |
| url | https://arxiv.org/abs/2511.16968 |