Order polarities

Abstract

Abstract We define an order polarity to be a polarity $(X,Y,{\operatorname{R}})$ where $X$ and $Y$ are partially ordered, and we define an extension polarity to be a triple $(e_X,e_Y,{\operatorname{R}})$ such that $e_X:P\to X$ and $e_Y:P\to Y$ are poset extensions and $(X,Y,{\operatorname{R}})$ is an order polarity. We define a hierarchy of increasingly strong coherence conditions for extension polarities, each equivalent to the existence of a preorder structure on $X\cup Y$ such that the natural embeddings, $\iota _X$ and $\iota _Y$, of $X$ and $Y$, respectively, into $X\cup Y$ preserve the order structures of $X$ and $Y$ in increasingly strict ways. We define a Galois polarity to be an extension polarity satisfying the strongest of these coherence conditions and where $e_X$ and $e_Y$ are meet- and join-extensions, respectively. We show that for such polarities the corresponding preorder on $X\cup Y$ is unique. We define morphisms for polarities, providing the class of Galois polarities with the structure of a category, and we define an adjunction between this category and the category of $\varDelta _1$-completions and appropriate homomorphisms.