Spectral duality for some modal and residuated groupoid expansions of De Morgan algebras

Abstract

Stone demonstrated that the category $\mathbf{DLAT}{\mathbf{T}}_{0,1}$ of bounded distributive lattices is dually equivalent to the category $\mathbf{Spec}$ of spectral spaces and Priestley showed that $\mathbf{DLat}{\mathbf{t}}_{0,1}$ is dually equivalent to the category $\mathbf{Priest}$ of Priestley spaces so that $\mathbf{Spec}$ is equivalent $\mathbf{Priest}$. Cornish strengthened this by showing that $\mathbf{Spec}$ and $\mathbf{Priest}$ are in fact isomorphic. In this study, we investigate the duality theory of various lattice expansions of certain bounded distributive lattice-ordered algebras, known as De Morgan algebras. In particular we obtain spectral duality results for the category $S4DM$ of De Morgan algebras equipped with a closure operator, which we call S4 De Morgan algebras, as well as for the category $\mathbf{DMGrp}$ of De Morgan groupoids. This is achieved by an appropriate adaptation of Bimbó's Priestley-style duality for general De Morgan algebras together with Urquhart's Priestley-style duality for relevance algebras under the isomorphism between $\mathbf{Priest}$ and $\mathbf{Spec}$.