The subject of the paper is suggested by G. Janelidze and motivated by his earlier result giving a positive answer to the question posed by S. MacLane whether the Galois theory of homogeneous linear ordinary differential equations over a differential field (which is Kolchin–Ritt theory and an algebraic version of Picard–Vessiot theory) can be obtained as a particular case of G. Janelidze’s Galois theory in categories. One ground category in the Galois structure involved in this theory is dual to the category of commutative rings with unit, and another one is dual to the category of commutative differential rings with unit. In the present paper, we apply the general categorical construction, the particular case of which gives this Galois structure, by replacing “commutative rings with unit” by algebras from any variety \mathscr{V} of universal algebras satisfying the amalgamation property and a certain condition (of the syntactical nature) for elements of amalgamated free products which was introduced earlier, and replacing “commutative differential rings with unit” by \mathscr{V}-algebras equipped with additional unary operations which satisfy some special identities to construct a new Galois structure. It is proved that this Galois structure is admissible. Moreover, normal extensions with respect to it are characterized in the case where \mathscr{V} is any of the following varieties: abelian groups, loops and quasigroups.

中文翻译：

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对某些泛代数种类的对偶范畴的可接受伽罗瓦结构

这篇论文的主题是由 G. Janelidze 提出的，并受到他早期结果的启发，该结果对 S. MacLane 提出的问题给出了肯定的答案，即微分域上齐次线性常微分方程的伽罗瓦理论（Kolchin-Ritt 理论）和 Picard-Vessiot 理论的代数版本）可以作为 G. Janelidze 的 Galois 理论在范畴中的一个特例获得。该理论所涉及的伽罗瓦结构中的一个基范畴是对偶的有单位的交换环范畴，另一个是对偶的有单位的交换微分环范畴。在本文中，我们应用一般分类构造，其特殊情况给出了这种伽罗瓦结构，通过用满足合并性质和特定条件（句法性质）的通用代数的任何变体 \mathscr{V} 中的代数替换“带单位的交换环”，并替换“可交换的自由乘积”带有单位的微分环”由 \mathscr{V}-代数配备了额外的一元运算，满足一些特殊的恒等式来构造一个新的伽罗瓦结构。证明了这种伽罗瓦结构是可接受的。此外，在 \mathscr{V} 是以下任何变体的情况下，关于它的正常扩展的特征是：阿贝尔群、环和拟群。并用\mathscr{V}-代数替换“带单位的可交换微分环”，该代数配备了额外的一元运算，满足一些特殊的恒等式以构建新的伽罗瓦结构。证明了这种伽罗瓦结构是可接受的。此外，在 \mathscr{V} 是以下任何变体的情况下，关于它的正常扩展的特征是：阿贝尔群、环和拟群。并用配备额外一元运算的 \mathscr{V}-代数替换“带单位的可交换微分环”，以构建新的伽罗瓦结构。证明了这种伽罗瓦结构是可接受的。此外，在 \mathscr{V} 是以下任何变体的情况下，关于它的正常扩展的特征是：阿贝尔群、环和拟群。