揭示Eilenberg型对应：（有限）代数的Birkhoff定理+对偶
Unveiling Eilenberg-type Correspondences: Birkhoff's Theorem for (finite) Algebras + Duality
摘要：
本文的目的是证明：Eilenberg型对应=（有限）代数的Birkhoff定理+对偶。我们考虑代数单子T上的范畴D，我们研究（伪）品种的T-代数。代数的伪簇在文献中也称为有限代数的簇。两个著名的定理，其特点是品种和伪品种的代数发挥了重要作用，在这里：伯克霍夫定理和伯克霍夫定理有限代数，后者也被称为赖特曼定理。我们证明，在温和的假设下，一个明确的版本Birkhoff定理（有限）代数建立一个一对一的对应关系（伪）品种的T-代数和（伪）方程T-理论。现在，如果C是与D对偶的范畴，而B是C上T的对偶的共单子，我们得到（伪）方程T-理论与它们的对偶（伪）余方程B-理论之间的一一对应。（伪）余等式B-理论的特殊例子已经在语言理论中以“语言的多样性”的名义进行了研究，以建立艾伦伯格型对应。总之，我们得到了T-代数的（伪）簇与（伪）余方程B-理论之间的一一对应，这正是Eilenberg型对应的性质。
英文摘要：
The purpose of the present paper is to show that: Eilenberg-type correspondences = Birkhoff's theorem for (finite) algebras + duality. We consider algebras for a monad T on a category D and we study (pseudo)varieties of T-algebras. Pseudovarieties of algebras are also known in the literature as varieties of finite algebras. Two well-known theorems that characterize varieties and pseudovarieties of algebras play an important role here: Birkhoff's theorem and Birkhoff's theorem for finite algebras, the latter also known as Reiterman's theorem. We prove, under mild assumptions, a categorical version of Birkhoff's theorem for (finite) algebras to establish a one-to-one correspondence between (pseudo)varieties of T-algebras and (pseudo)equational T-theories. Now, if C is a category that is dual to D and B is the comonad on C that is the dual of T, we get a one-to-one correspondence between (pseudo)equational T-theories and their dual, (pseudo)coequational B-theories. Particular instances of (pseudo)coequational B-theories have been already studied in language theory under the name of "varieties of languages" to establish Eilenberg-type correspondences. All in all, we get a one-to-one correspondence between (pseudo)varieties of T-algebras and (pseudo)coequational B-theories, which will be shown to be exactly the nature of Eilenberg-type correspondences.
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