Monotone maps and Galois connections

Let $(P, \subseteq)$ and $(Q,\subseteq)$ be partially ordered sets and $f: (P, \subseteq) \to (Q,\subseteq)$ a function between them. Recall that a function is a one-to-one map between the lattices $Part(P)^d$ and $\mathcal{P}(Q)$. These are the lattices of subobjects and quotients of the topoi of sets. In the special case of functions between partial orders the equivalence classes and images of the function produce suborders. We will consider these suborders in the case of monotone maps in general and Galois connections in particular.

Maximal and minimal representatives:
The function $f : (P,\subseteq) \to (Q, \subseteq)$ induces an equivalence relation $x =_f y \Leftrightarrow f(x) = f(y)$. If the equivalence classes of $=_f$ are upper bounded suborders, then there is a maximal representative input that produces any given output. Dually, if the equivalence classes of $=_f$ are lower bounded suborders, then minimal representatives exist. It is also possible that both or neither exist. The existence of maximal and minimal representatives can be represented symbolically:

• Maximal representatives: $(F(a) = b) \Rightarrow (a \subseteq G(b))$
• Minimal representatives: $(F(a) = b) \Rightarrow (G(b) \subseteq a)$

Maximization produces an increasing idempotent action and minimization produces a decreasing idempotent action. For a given map between posets for which either process is possible, maximization and minimization don't necessarily have to be monotone. In the case which they are such as in a Galois connection then they form closure and interior operators.

Image suborders:
The image of $f : (P, \subseteq) \to (Q,\subseteq)$ is a suborder of $(Q, \subseteq)$. We can therefore classify monotone maps between partial orders based upon the properties of their image suborders. We say that a suborder is a Moore suborder if it has a closure operation and a Comoore suborder if it has an interior operation. These are the two special cases of interest in Galois connections.

Suppose that $f : (P, \subseteq) \to (Q,\subseteq)$ is a monotone map and we have $(F(a) \subseteq b) \Rightarrow (a \subseteq G(b))$. Then by monotonicity we have that $F(a) \subseteq F(G(b))$. Which proves that the $F$ image of $G(b)$ is greater then any other which is less then $b$. This means there must be on the images of $f$. Dually, if we have the same condition in the other direction there must be a closure operator on the images of $f$.

Overview:
There are three types of map associated to Galois connections: lower adjoints, upper adjoints, and polarities. Upper and lower adjoints are monotone and polarities are antitone. We can use the results proved thus far to describe the images and equivalence classes of each type of map:

• Lower adjoints:
  • Upper bounded equivalence classes
  • An interior operator of images
• Upper adjoints:
  • Lower bounded equivalence classes
  • A closure operator of images
• Polarities:
  • Upper bounded equivalence classes
  • A closure operator of images

A Galois connection is from a lower adjoint by taking the maximal representative of the largest image less then a given output value. For an upper adjoint it is determined by taking the minimal representative of the smallest image greater then a output value. Finally, for a polarity it is determined by the maximal representative of the closure of an output value. In this way, when Galois connections exist they are uniquely determined by each map.

For example, in algebraic geometry we saw that the antitone map $V: \wp(R[x,y,z,...]) \to \wp(\mathbb{A}^n)$ from any polynomial system to its algebraic set has upper bounded equivalence classes and its image has a closure operator. Therefore, V is a polarity in an antitone Galois connection. There are stronger properties, like that the images form a cotopology (the Zariski cotopology) and in the case of an algebraically closed field the maximal representatives are radical ideals.

As a function is a one-to-one map between its equivalence classes and its image, in order to construct a one-to-one restriction mapping of a function it is only necessary to get representatives of each equivalence class. In both types of Galois connection this can be achieved by selecting minimal and maximal representatives of a map. In the case of algebraic geometry, by Hilbert's nullstellensatz there is a one to one mapping between radical ideals and algebraic sets of a polynomial ring over an algebraically closed field.

Suprema and infima:
The purpose of this post is to compile all the relevant aspects of the monotone maps of a Galois connection, rather then considering the connections themselves. There is one more property worth considering and that is the relationship between maps and suprema/infima:

• Lower adjoints preserve suprema
• Upper adjoints preserve infima

Order theory and category theory comparison:
Almost everything in category theory is a slight variant of something older in order theory. One reason to consider Galois connections then, is their relationship to category theoretic adjoints. In that case, the most important property is that lower adjoints preserve colimits and upper adjoints preserve limits. An example of relevance to algebraic-geometry is the tensor-hom adjunction. There we see that the tensor product is a lower adjoint and so it preserves colimts and the hom is an upper adjoint so it preserves limits. The limit/colimit preserving conditions for adjoint functors correspond to the suprema/infima preserving conditions of Galois connections.