Order polarities
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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→ X and e_Y:P→ 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∪ Y such that the natural embeddings, ι_X and ι_Y, of X and Y, respectively, into X∪ 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∪ Y such that ι_X and ι_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 Δ_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.
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