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 $G:2^B\to 2^A$:

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

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:

$S\subset S^{\prime }⟹F(S^{\prime })\subset F(S)$

and:

$T\subset T^{\prime }⟹G(T^{\prime })\subset G(T)$

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

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

and:

$S\subset S^{\prime }⟹G(F(S))\subset G(F(S^{\prime }))$

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