2024/06/11 by John Harding, Joseph McDonald, Harding, John +3 · 4 citations
Arts and Humanities · #06B23 06E15 #06C15 #Classical Philosophy and Thought #FOS: Mathematics #FOS: Physical sciences #Logic (math.LO) #Quantum Physics (quant-ph)
paper · pdf · doi:10.48550/arxiv.2406.06917
openalex publication_date 2024/06/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.