Dual Plo\v{s}\v{c}ica spaces of ortholattices

Abstract

We describe digraphs with topology which give dual representations of ortholattices. This is done via so-called dual Plo\v{s}\v{c}ica spaces of lattices. First, we improve the definition of Plo\v{s}\v{c}ica spaces from an earlier paper to give a straight and natural generalisation of the total order disconnectedness of Priestley spaces. Then we define the dual space of a general ortholattice as the dual Plo\v{s}\v{c}ica space of the lattice-reduct of the ortholattice equipped with a map representing the orthocomplement operation. We introduce an abstract ortho-Plo\v{s}\v{c}ica space capturing the properties of the dual space of an ortholattice, and we present dual representation theorems between general ortholattices and the ortho-Plo\v{s}\v{c}ica spaces. We illustrate our dual representations by examples.