有限Cartesian闭品种与半群作用
Finitary Cartesian closed varieties and semigroup actions
AI总结：
本文通过布尔左限制单oid的结构，将有限Cartesian闭品种与半群作用联系起来，揭示了两者之间的等价关系。
摘要：
我们借鉴Richard Garner的一些想法。设M是一个单oid，B是一个布尔代数。一个'匹配对'[B|M]由B和M以及一些相互作用组成。Garner证明了每个这样的匹配对确定（我们称之为）一个布尔左限制单oid S=S[B|M]。在本文中，我们表明[B|M]-set（稍后定义）的数据可以通过某种由S作用的方式编码。这意味着[B|M]-sets的范畴等价于S-作用的范畴。我们得出结论，Garner的工作表明，每个非退化的有限Cartesian闭品种等价于一个特殊的S-作用范畴，其中S是一个布尔左限制单oid。
英文摘要：
We build on some ideas of Richard Garner. Let $M$ be a monoid and $B$ a Boolean algebra. A `matched pair' $[B|M]$ consists of $B$ and $M$ and some mutual interactions. Garner showed that every such matched pair determines (what we shall call) a Boolean left restriction monoid $S = S[B|M]$. In this paper, we show that the data of a $[B|M]$-set (defined later) may be encoded by means of a certain kind of action by $S$. This means that the category $[B|M]$-{\bf sets} is equivalent to a category of {\bf $S$-actions}. We deduce, as a result of Garner's work, that every non-degenerate finitary Cartesian closed variety is equivalent to a special category of $S$-actions where $S$ is a Boolean left restriction monoid.
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