Notes on foundations. II. On Galois connections.
G. Y. Rainich · Notre Dame Journal of Formal Logic · 1962
Note II. On Galois ConnectionsWe deal here with partially ordered sets.In such a set a relation of order is defined which possesses the usual properties except that given two elements it is not necessarily true that one is higher than the other or that they are equally high.As an example we may consider the set of subsets of a given set; given two such subsets it may happen that one includes the other or that each includes the other but, in general neither of these relations is true.We say in a case like this that we have partial order β by inclusion*.Although in a sense all cases of partial order may be reduced to this it is convenient to consider other situations independently.Examples of some other cases appear in what follows.Given two partially ordered sets which are in correspondence it may be that their orderings are related to each other.A classical example occurs in the theory of algebraic equations.As one set we may consider the set of rational functions of the roots of a polynomial-these functions are called natural irrationalities.One such natural irrationality may be called higher than another if the second is a rational function of the first but not vice versa.Under this definition the natural irrationalities constitute a partially ordered set.On the other hand, we may consider the groups of permutations of the roots of our polynomial.If in connection with every natural irrationality we consider the group of permutations of the roots under which the natural irrationality is invariant we have established a correspondence between the two partially ordered sets, the set of natural irrationalities and the set of groups of permutations ordered by inclusion.It is true that when a natural irrationality ce is a rational function of β the permutations that do not affect β would not affect oe so that when a is higher than β its group includes the group of β.There is then a connection between the partial orders of the two sets.This forms the basis of the Lagrange-Galois theory.If there is a correspondence between two partially ordered sets such that whenever an element α of the first set is higher than an element β it is true that the corresponding elements of the second set are in the same order relation we say that the partially ordered sets are isotaxic; the term "Galois connected* is used with about the same meaning.The two sets considered above in connection with an algebraic equation are isotaxic.