关于藤原刚的态射与变换（作为二维多序一般代数的具体化及其在多序克隆与代数理论等价性中的应用）
On the Morphisms and Transformations of Tsuyoshi Fujiwara (as a concretion of a bidimensional many-sorted general algebra and its application to the equivalence between many-sorted clones and algebraic theories)
摘要：
对于单序代数，Fujiwara通过基本映射公式族的概念，定义了一个态射的概念，它推广了代数间同态的一般概念，并在基本映射公式族上定义了一个等价关系，即共轭，它对应于代数的内同构关系。本文将Fujiwara关于态射的理论推广到多类代数，定义了多类签名之间的多导子概念，它赋予基本类、字和形式运算、导项族，并将标准签名态射、Fujiwara的基本映射公式和Goguen-Thatcher-Wagner的导子都包含在多导子之下.然后，通过Bénabou代数之间的同态，这是Bénabou的有限多序代数理论的代数对应，我们定义了复合的多导子，从我们得到一个相应的范畴，同构于Kleisli范畴的单子上的标准范畴的多序签名。其次，通过定义多导子之间的转换概念，我们赋予了多分类签名和多导子范畴2-范畴的结构。由此得到了一个导出的2-范畴的多序规范，证明了Hall代数和Bénabou代数的多序规范的等价性，并导出了Hall代数和Bénabou代数的范畴的等价性.此外，通过定义广义多序项的相应范畴，证明了这些广义多序项在多序代数中的实现在多导子下是不变的，并且与多导子之间的变换是相容的，由此得到了2-机构的一个新概念，它本身是Goguen和Burstall的机构概念的严格推广.
英文摘要：
For single-sorted algebras, Fujiwara defined, through the concept of family of basic mapping-formulas, a notion of morphism which generalizes the ordinary notion of homomorphism between algebras and an equivalence relation, the conjugation, on the families of basic mapping-formulas, which corresponds to the relation of inner isomorphism for algebras. In this paper we extend the theory of Fujiwara about morphisms to the many-sorted algebras, by defining the concept of polyderivor between many-sorted signatures, which assigns to basic sorts, words and to formal operations, families of derived terms, and under which the standard signature morphisms, the basic mapping-formulas of Fujiwara, and the derivors of Goguen-Thatcher-Wagner are subsumed. Then, by means of the homomorphisms between Bénabou algebras, which are the algebraic counterpart of the finitary many-sorted algebraic theories of Bénabou, we define the composition of polyderivors from which we get a corresponding category, isomorphic to the category of Kleisli for a monad on the standard category of many-sorted signatures. Next, by defining the notion of transformation between polyderivors, we endow the category of many-sorted signatures and polyderivors with a structure of 2-category. From this we get a derived 2-category of many-sorted specifications in which we prove the equivalence of the many-sorted specifications of Hall and Bénabou, and deduce the equivalence of the categories of Hall and Bénabou algebras. Besides, by defining corresponding categories of generalized many-sorted terms, we prove that the realization of these terms in the many-sorted algebras is invariant under polyderivors and compatible with the transformations between polyderivors, and from this we get an example, among others, of the new concept of 2-institution, itself an strict generalization of that of institution by Goguen and Burstall.
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