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Abstract Galois theory and endotheory. I. (English) Zbl 0632.08001

Let K be a field, G a finite group of automorphisms of K and \(\rho\subset K\times G\) the relation defined by \[ k\rho g\Leftrightarrow g(k)=k. \] The fundamental theorem of classical Galois theory gives the following description of the closure operators on subsets of K and G induced by the Galois connection corresponding to \(\rho\) : a subset K’\(\subset K\) is closed iff it is a subfield containing \(K^ G\), and a subset G’\(\subset G\) is closed iff it is a subgroup.
The author gives another version of Galois theory: Let E be a set, X a set with card(X)\(\geq card(E)\), R the set of X-ary relations on E and S the permutation group of E; instead of \(\rho\) consider the relation \(\rho^*\subset R\times S\) defined by \[ r\rho^*s\Leftrightarrow (s\quad preserves\quad r), \] then: a subset R’\(\subset R\) is closed iff it is closed with respect to the so-called “fundamental operations” (they are infinite generalizations of the operations of predicate calculus), and a subset S’\(\subset S\) is closed iff it is a subgroup. The “Endo-theory” is the analogous description in the case when we consider the monoid of self-mappings of E instead of S.
The author considers examples; in particular, he observes that the fundamental theorem of the classical Galois theory (even infinite) can be deduced from his results, but the deduction is “quite complicated”.
Remark. The author notes that he has found the abstract Galois theory during the summer vacation of 1935 and gives other historical remarks [see also M. Krasner, Ann. Sci. Univ. Clermont 60, Math. 13, 87-91 (1976; Zbl 0376.08001) and other previous papers of the author].
As it seems to me, unfortunately his famous results have not been estimated so far.
Reviewer: G.Dzhanelidze

MSC:

08A05 Structure theory of algebraic structures
20M20 Semigroups of transformations, relations, partitions, etc.
12F10 Separable extensions, Galois theory
20B27 Infinite automorphism groups
08A65 Infinitary algebras
08A35 Automorphisms and endomorphisms of algebraic structures
06A15 Galois correspondences, closure operators (in relation to ordered sets)

Citations:

Zbl 0376.08001
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