秩序极性
Order polarities
摘要：
我们将序极性定义为极性$(X,Y,R)$，其中$X$和$Y$是偏序的，并且我们将扩展极性定义为三元组$(e_X,e_Y,R)$，使得$e_X:P\to X$和$e_Y:P\to Y$是偏序集扩展，并且$(X,Y,R)$是序极性。我们定义了一个层次结构越来越强的连贯性条件的扩展极性，每个相当于存在一个前序结构的$X\cup Y$，这样的自然嵌入，$ι_X$和$ι_Y$，$X$和$Y$，分别到$X\cup Y$保持$X$和$Y$的顺序结构，在越来越严格的方式。我们定义的伽罗瓦极性是一个扩展极性$e_X$和$e_Y$满足和加入扩展分别，我们表明，这样的极性有一个独特的前序$X\cup Y$，使$ι_X$和$ι_Y$满足特别强的保存性能。我们定义态射的极性，提供类的伽罗瓦极性的一个类别的结构，我们定义了这个类别和类别之间的附加$Δ_1$ -完成和适当的同态。我们形式化的扩展极性理论，并证明了一个对偶原理的效果，如果一个声明是真的所有扩展极性，那么也必须是它的对偶声明。
英文摘要：
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, $ι_X$ and $ι_Y$, 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 $ι_X$ 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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