Order polarities

Abstract

We define an order polarity to be a polarity $(X,Y,R)$ where $X$ and $Y$ are partially ordered, and we define an extension polarity to be a triple $(e_X,e_Y,R)$ such that $e_X:P\to X$ and $e_Y:P\to Y$ are poset extensions and $(X,Y,R)$ is an order polarity. We define a hierarchy of increasingly strong coherence conditions for extension polarities, each equivalent to the existence of a pre-order 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 where $e_X$ and $e_Y$ are meet- and join-extensions respectively, and we show that for such polarities there is a unique pre-order on $X\cup Y$ such that $\iota_X$ and $\iota_Y$ satisfy particularly strong preservation properties. 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 $\Delta_1$-completions and appropriate homomorphisms. We formalize the theory of extension polarities and prove a duality principle to the effect that if a statement is true for all extension polarities then so too must be its dual statement.