Monadic ortholattices: completions and duality
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| Format: | Preprint |
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2024
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| _version_ | 1866929381134303232 |
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| author | Harding, John McDonald, Joseph Peinado, Miguel |
| author_facet | Harding, John McDonald, Joseph Peinado, Miguel |
| contents | We show that the variety of monadic ortholattices is closed under MacNeille and canonical completions. In each case, the completion of $L$ is obtained by forming an associated dual space $X$ that is a monadic orthoframe. This is a set with an orthogonality relation and an additional binary relation satisfying certain conditions. For the MacNeille completion, $X$ is formed from the non-zero elements of $L$, and for the canonical completion, $X$ is formed from the proper filters of $L$. The corresponding completion of $L$ is then obtained as the ortholattice of bi-orthogonally closed subsets of $X$ with an additional operation defined through the binary relation of $X$.
With the introduction of a suitable topology on an orthoframe, as was done by Goldblatt and Bimbó, we obtain a dual adjunction between the categories of monadic ortholattices and monadic orthospaces. A restriction of this dual adjunction provides a dual equivalence. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_06917 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Monadic ortholattices: completions and duality Harding, John McDonald, Joseph Peinado, Miguel Logic Quantum Physics 06C15, 06B23 06E15 We show that the variety of monadic ortholattices is closed under MacNeille and canonical completions. In each case, the completion of $L$ is obtained by forming an associated dual space $X$ that is a monadic orthoframe. This is a set with an orthogonality relation and an additional binary relation satisfying certain conditions. For the MacNeille completion, $X$ is formed from the non-zero elements of $L$, and for the canonical completion, $X$ is formed from the proper filters of $L$. The corresponding completion of $L$ is then obtained as the ortholattice of bi-orthogonally closed subsets of $X$ with an additional operation defined through the binary relation of $X$. With the introduction of a suitable topology on an orthoframe, as was done by Goldblatt and Bimbó, we obtain a dual adjunction between the categories of monadic ortholattices and monadic orthospaces. A restriction of this dual adjunction provides a dual equivalence. |
| title | Monadic ortholattices: completions and duality |
| topic | Logic Quantum Physics 06C15, 06B23 06E15 |
| url | https://arxiv.org/abs/2406.06917 |