Galois correspondence induced by a binary relation

This article is about a general term. A list of important particular cases (instances) is available at Category:Galois correspondences induced by binary relations

Definition

Let $A$ and $B$ be sets. Suppose $R\subset A\times B$ i.e. $R$ is a binary relation between $A$ and $B$ . $R$ induces a Galois correspondence between subsets of $A$ and subsets of $B$ , as follows. We get two maps, $F:2^{A}\to 2^{B}$ and $G:2^{B}\to 2^{A}$ :

$F(S)=\{b\in B|aRb\ \forall \ a\in S\}$
$G(T)=\{a\in A|aRb\ \forall \ b\in T\}$

Key facts

$F$ and $G$ are both reverse-monotone. In category-theoretic jargon, $F$ and $G$ are contravariant functors between the categories of subsets of $A$ and $B$ , with morphisms being inclusions.

In symbols:

$S\subset S'\implies F(S')\subset F(S)$

and:

$T\subset T'\implies G(T')\subset G(T)$

$F\circ G$ and $G\circ F$ are both monotone, and ascendant. In other words:

$S\subset G(F(S))$

and:

$S\subset S'\implies G(F(S))\subset G(F(S'))$

Similarly for $F\circ G$

$G\circ F\circ G=G$ and $F\circ G\circ F=F$ . In other words, going back and forth thrice has the same effect as going once. This follows easily from the last two facts.

The operator $F\circ G$ defines a closure operator on $B$ , and the operator $G\circ F$ defines a closure operator on $A$ . By closure operator is meant a monotone ascendant operator. In particular, the subsets which are fixed under this operator, which are called the closed sets, include the whole set, and are closed under taking arbitrary intersections

In general, the closed sets on either side under a Galois correspondence, need not give a topology. The problem is that the empty set may not be closed, and a finite union of closed sets need not be closed. There are special circumstances under which we do get a topology, for instance the spectrum of a commutative unital ring or the max-spectrum of a commutative unital ring.

Related notions

Effect on Galois correspondence of restricting binary relation to a subset