笛卡尔闭簇I：分类定理
Cartesian closed varieties I: the classification theorem
摘要：
1990年，Johnstone给出了方程理论的一个句法特征，其相关的变量是Carnival封闭的。在这些理论中，所有的一元理论--其模型是由幺半群M赋予作用的集合--和所有的超仿射理论--其模型是由布尔代数B赋予作用的集合。我们改进Johnstone的结果表明，一个方程的理论是carnival封闭的，只是当它的操作有一个独特的超仿射一元分解。由此可以得出，任何非退化的卡氏闭簇都是由幺半群M和布尔代数B装配有相容作用的集合的簇;这是标题的分类定理。
英文摘要：
In 1990, Johnstone gave a syntactic characterisation of the equational theories whose associated varieties are cartesian closed. Among such theories are all unary theories -- whose models are sets equipped with an action by a monoid M -- and all hyperaffine theories -- whose models are sets with an action by a Boolean algebra B. We improve on Johnstone's result by showing that an equational theory is cartesian closed just when its operations have a unique hyperaffine-unary decomposition. It follows that any non-degenerate cartesian closed variety is a variety of sets equipped with compatible actions by a monoid M and a Boolean algebra B; this is the classification theorem of the title.
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