查看“︁对偶 (数学)”︁的源代码

在[[数学]]领域中，'''对偶'''一般来说是以一对一的方式，常常（但并不总是）通过某个[[对合]]算子，把一种概念、公理或数学结构转化为另一种概念、公理或数学结构：如果''A''的对偶是''B''，那么''B''的对偶是''A''。由于对合有时候会存在[[不动点]]，因此''A''的对偶有时候会是''A''自身。比如[[射影几何]]中的[[笛沙格定理]]，即是在这一意义下的自对偶。

''对偶''在数学背景当中具有很多种意义，而且，尽管它是“现代数学中极为普遍且重要的概念（a very pervasive and important concept in (modern) mathematics）”<ref>{{harvnb|Kostrikin|2001}}</ref>并且是“在数学几乎每一个分支中都会出现的重要的一般性主题（an important general theme that has manifestations in almost every area of mathematics）”<ref name="PCM187L">{{harvnb|Gowers|2008|loc=p.&nbsp;187, col.&nbsp;1}}</ref>，但仍然没有一个能把对偶的所有概念统一起来的普适定义。<ref name="PCM187L"/>

在两类对象之间的对偶很多都和[[配对]]（pairing），也就是把一类对象和另一类对象映射到某一族标量上的双线性函数相对应。例如，线性代数的对偶对应着把线性空间中的向量对双线性映射到标量上，[[广义函数]]及其相关的[[试验函数]]也对应着一个配对且在该配对中可用试验函数来对广义函数进行积分，[[庞加莱对偶]]从给定流形的子流形之间的配对的角度看同样也对应着[[交数]]。<ref name="PCM189R">{{harvnb|Gowers|2008|loc=p.&nbsp;189, col.&nbsp;2}}</ref>

==序逆对偶==
一种特别简单的对偶形式来自于[[序理论]]。偏序关系''P'' = (''X'', ≤)的对偶是由同一偏序集组成但关系相反的偏序关系''P<sup>d</sup>''。我们比较熟悉的对偶偏序的例子有：

* 任何集合簇上的子集和超集关系<math>\subset</math>和<math>\supset</math>；

* [[整数]]上的''因数''和''倍数''关系；

* 人类集合上的''后代''和''祖先''关系。

为某一偏序''P''定义的概念会对应到对偶偏序集''P<sup>d</sup>''的''对偶概念''上。例如，''P''的[[极小元]]对应于''P<sup>d</sup>''的[[极大元]]：极小和极大是序理论中的对偶概念。序理论中的其他对偶概念还包括[[上界和下界]]、[[上闭集合]]和[[下闭集合]]、[[理想 (序理论)|理想]]和[[滤子 (数学)|滤子]]。

一种特殊的序逆对偶存在于某个集合''S''的幂集合中：若<math>\bar A=S\setminus A</math>表示[[补集]]，则<math>A\subset B</math>当且仅当<math>\bar B\subset \bar A</math>。在拓扑学中，[[开集]]和[[闭集]]是对偶概念：开集的补是闭的，反之亦然。在[[拟阵]]论中，某个给定拟阵的独立集合的补集簇形成另一个拟阵，称作对偶拟阵。在[[逻辑]]中，我们可以把非量化公式中变量的[[成真赋值]]表示为对该赋值为真的变量集合。成真赋值满足该公式当且仅当该成真赋值的补满足该公式的[[德摩根定律]]。逻辑中的全称量词和存在量词也是类似的对偶。

偏序可以解释为[[范畴 (数学)|范畴]]，在该范畴中存在从''x''到''y''的arrow当且仅当偏序中有''x''&nbsp;≤&nbsp;''y''。偏序的序逆对偶可扩展为[[对偶范畴]]的概念，即由给定范畴中所有arrow的逆所组成的范畴。后面将要描述的很多具体的对偶都是在此意义下的范畴的对偶。

==维逆对偶==
[[File:Dual Cube-Octahedron.svg|thumb|200px|The features of the cube and its dual octahedron correspond one-for-one with dimensions reversed.]]
存在着很多种不同但互相联系的在同一类几何或拓扑对象之间的对偶，不过具有对偶关系的对象在特征维数上是相反的。这方面的经典例子是[[柏拉圖立體|正多面体]]的对偶，其中立方体和正八面体形成了一个对偶配对，正十二面体和正二十面体形成了另一个对偶配对，而正四面体是'''自对偶'''的。任何一種這類多面體的[[對偶多面體]]可作為主要多面體每一面中心點的[[凸包]]。

==相关条目==
* [[倒易点阵]]
* [[伴隨函子]]

==注释==
{{reflist|2}}

==參考資料==
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[[Category:對偶理論|*]]

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