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Admissible Galois Structures on the categories dual to some varieties of universal algebras
- Source :
- Georgian Mathematical Journal. 28:651-664
- Publication Year :
- 2020
- Publisher :
- Walter de Gruyter GmbH, 2020.
-
Abstract
- 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 V \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 V \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 V \mathscr{V} is any of the following varieties: abelian groups, loops and quasigroups.
- Subjects :
- Pure mathematics
General Mathematics
010102 general mathematics
Galois theory
Commutative ring
Amalgamation property
01 natural sciences
010101 applied mathematics
Differential algebra
0101 mathematics
Variety (universal algebra)
Abelian group
Commutative property
Unit (ring theory)
Mathematics
Subjects
Details
- ISSN :
- 15729176 and 1072947X
- Volume :
- 28
- Database :
- OpenAIRE
- Journal :
- Georgian Mathematical Journal
- Accession number :
- edsair.doi...........21d675d4b5f3bec2a958fe6c26839bc3