Monadic ortholattices: completions and duality

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 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.