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<p>We show that the variety of monadic ortholattices is closed under MacNeille and canonical completions. In each case, the completion of <i>L</i> is obtained by forming an associated dual space <i>X</i> 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, <i>X</i> is formed from the non-zero elements of <i>L</i>, and for the canonical completion, <i>X</i> is formed from the proper filters of <i>L</i>. The corresponding completion of <i>L</i> is then obtained as the ortholattice of bi-orthogonally closed subsets of <i>X</i> with an additional operation defined through the binary relation of <i>X</i>. 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.</p>

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Monadic ortholattices: completions and duality

  • 期刊论文

  • 发表日期：

    2025年4月3日

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  • John Harding,
  • Joseph McDonald,
  • Miguel Peinado

摘要

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.