Monadic ortholattices: completions and duality

TL;DR

This paper investigates the structure of monadic ortholattices, demonstrating their closure under certain completions and establishing a duality with monadic orthospaces through topological and categorical methods.

Contribution

It introduces a duality framework for monadic ortholattices using orthoframes and completions, extending the understanding of their algebraic and topological properties.

Findings

Monadic ortholattices are closed under MacNeille and canonical completions.

A dual adjunction and a dual equivalence are established between monadic ortholattices and monadic orthospaces.

Completion of lattices is characterized via bi-orthogonally closed subsets of associated dual spaces.

Abstract

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 Bimb\'o, we obtain a dual adjunction between the categories of monadic ortholattices and monadic orthospaces. A restriction of this dual adjunction provides a dual equivalence.