Remarks on abstract Galois theory

Abstract

This paper is a historical companion to a previous one, in which it was studied the so-called abstract Galois theory as formulated by the Portuguese mathematician José Sebastião e Silva (see da Costa, Rodrigues (2007)). Our purpose is to present some applications of abstract Galois theory to higher-order model theory, to discuss Silva's notion of expressibility and to outline a classical Galois theory that can be obtained inside the two versions of the abstract theory, those of Mark Krasner and of Silva. Some comments are made on the universal theory of (set-theoretic) structures.

Galois Theory of Algorithms

“…Yet another way of viewing this work is an application and a variation of some ideas from universal algebra and model theory. In the literature, there is some discussion of Galois theory for arbitrary universal algebraic structures ( [4] section II.6) and and m-theoretic structures ( [6,5,8,16].) In broad philosophical terms, following the work of Galois and Klein's erlangen program, an object can be defined by looking at its symmetries.…”