Galois Connections. Part 3.

Let (P,≤P) and (Q,≤Q) be posets, and consider two set functions ∗ ∶ P ⇄ Q ∶ ∗. We will denote these by p ↦ p ∗ and q ↦ q ∗ for all p ∈ P and q ∈ Q. This pair of functions is called a Galois connection if, for all p ∈ P and q ∈ Q, we have

p ≤ P q ∗ ⇐⇒ q ≤ Q p  ∗

Let ∗ ∶ P ⇄ Q ∶ ∗ be a Galois connection. For all elements x of P or Q we will use the notations x ∗ ∗ ∶= (x ∗)∗ and x ∗ ∗ ∗ ∶= (x ∗ ∗)∗.

(1) For all p ∈ P and q ∈ Q we have

p ≤ P p ∗ ∗ and q ≤ Q q ∗ ∗.

(2) For all elements p1, p2 ∈ P and q1, q2 ∈ Q we have

p1 ≤ P p2 ⇒ p ∗ 2 ≤ Q p ∗ 1 and q1 ≤ Q q2 ⇒ q2 ∗ ≤ P q1 ∗.

(3) For all elements p ∈ P and q ∈ Q we have

p ∗ ∗ ∗ = p ∗ and q ∗ ∗ ∗ = q ∗

Proof:

Since the definition of a Galois connection is symmetric in P and Q, we will simplify the proof by using the notation

x ≤ y ∗ ⇐⇒ y ≤ x ∗

for all elements x,y such that the inequalities make sense. To prove (1) note that for any element x we have x ∗ ≤ x ∗ by the reflexivity of partial order. Then from the definition of Galois connection we obtain,

(x ∗) ≤ (x) ∗ ⇒ (x) ≤ (x ∗) ∗ ⇒ x ≤ x ∗ ∗

To prove (2) consider elements x, y such that x ≤ y. From (1) and the transitivity of partial x ≤ y ≤ y ∗ ∗ ⇒ x ≤ y ∗ ∗. Then from the definition of Galois connection we obtain

(x) ≤ (y ∗) ∗ ⇒ (y ∗) ≤ (x) ∗ ⇒ y ∗ ≤ x ∗.

To prove (3) consider any element x. On the one hand, part (1) tells us that

(x ∗) ≤ (x ∗) ∗ ∗ ⇒ x ∗ ≤ x ∗ ∗ ∗.

On the other hand, part (1) tells us that x ≤ x ∗ ∗ and then part (2) says that

(x) ≤ (x ∗ ∗) ⇒ (x ∗ ∗) ∗ ≤ (x) ∗ ⇒ x ∗ ∗ ∗ ≤ x ∗

Finally, the antisymmetry of partial order says that x∗∗∗ = x∗, which we interpret as isomorphism of objects in the poset category. The following definition captures the essence of these three basic properties.

Definition of Closure in a Poset. Given a poset (P,≤), we say that a function cl ∶ P → P is a closure operator if it satisfies the following three properties:

(i) Extensive: ∀p ∈ P, p ≤ cl(p)

(ii) Monotone: ∀ p,q ∈ P, p ≤ q ⇒ cl(p) ≤ cl(q)

(iii) Idempotent: ∀ p ∈ P, cl(cl(p)) = p.

[Remark: If P = 2U is a Boolean lattice, and if the closure cl ∶ 2U → 2U also preserves finite unions, then we call it a Kuratowski closure. Kuratowski proved that such a closure is equivalent to a topology on the set U.]

If ∗ ∶ P → Q ∶ ∗ is a Galois connection, then the basic properties above immediately imply that the compositions ∗ ∗ ∶ P → P and ∗ ∗ ∶ Q → Q are closure operators.

Proof: Property (ii) follows from applying property (2) twice and property (iii) follows from applying to property (3).

Fundamental Theorem of Galois Connections: Any Galois connection ∗ ∶ P ⇄ Q ∶ ∗ determines two closure operators ∗ ∗ ∶ P → P and ∗ ∗ ∶ Q → Q. We will say that the element p ∈  P (resp. q ∈  Q) is ∗ ∗-closed if p∗ ∗ = p (resp. q∗ ∗ = q). Then the Galois connection restricts to an order-reversing bijection between the subposets of ∗ ∗-closed elements.

Proof: Let Q ∗ ⊆ P and P ∗ ⊆ Q denote the images of the functions ∗ ∶ Q → P and ∗ ∶ P  → Q, respectively. The restriction of the connection to these subsets defines an order-reversing bijection:

img_20170204_065156

Indeed, consider any p ∈ Q ∗, so that p = q ∗ for some q ∈ Q. Then by properties (1) and (3) of Galois connections we have

(p) ∗ ∗ = (q ∗) ∗ ∗ ⇒ p ∗ ∗ = q ∗ ∗ ∗ ⇒ p ∗ ∗ = q ∗ ⇒ p ∗ ∗ = p

Similarly, for all q ∈ P ∗ we have q ∗ ∗ = q. The bijections reverse order because of property (2).

Finally, note that Q ∗ and P ∗ are exactly the subsets of ∗ ∗-closed elements in P and Q, respectively. Indeed, we have seen above that every element of Q ∗ is ∗ ∗-closed. Conversely, if p ∈ P is ∗ ∗-closed then we have

p = p ∗ ∗ ⇒ p = (p ∗) ∗,

and it follows that p ∈ Q ∗. Similarly, every element of P ∗ is ∗ ∗-closed.

Thus, a Galois connection is something like a “loose bijection”. It’s not necessarily a bijection but it becomes one after we “tighten it up”. Sort of like tightening your shoelaces.

img_20170204_071135

The shaded subposets here consist of the ∗ ∗-closed elements. They are supposed to look (anti-) isomorphic. The unshaded parts of the posets get “tightened up” into the shaded subposets. Note that the top elements are ∗ ∗-closed. Indeed, property (2) tells us that 1P ≤ P ≤ 1p∗∗ and then from the universal property of the top element we have 1P** = 1P. Since the left hand side is always true, so is the right hand side. But then from the universal property of the top element in Q we conclude that 0P = 1Q. As a consequence of this, the arbitrary meet of ∗ ∗-closed elements (if it exists) is still ∗ ∗-closed. We will see, however, that the join of ∗ ∗-closed elements is not necessarily ∗ ∗-closed. And hence not all Galois connections induce topologies.

Galois connections between Boolean lattices have a particularly nice form, which is closely related to the universal quantifier ““. Galois Connections of Boolean Lattices. Let U,V be sets and let ∼ ⊆ U × V be any subset (called a relation) between U and V . As usual, we will write “u ∼ v” in place of the statement “(u,v) ∈ ∼“, and we read this as “u is related to v“. Then for all S ∈ 2U and T ∈ 2V we define,

S ∶= {v ∈ V ∶ ∀ s ∈ S, s ∼ v} ∈ 2V,

T ∶= {u ∈ U ∶ ∀ t ∈ T , u ∼ t} ∈ 2U

The pair of functions S ↦ S and T ↦ T is a Galois connection, ∼ ∶ 2U ⇄ 2V ∶ ∼.

To see this, note that ∀ subsets S ∈ 2U and T ∈ 2V we have

S ⊆ T ⇐⇒ ∀ s ∈ S, s ∈ T

⇐⇒ ∀ s ∈ S,∀ t ∈ T, s ∼ t

⇐⇒ ∀ t ∈ T, ∀ s ∈ S, s ∼ t

⇐⇒ ∀ t ∈ T, t ∈ S

⇐⇒ T ⊆ S.

Moreover, one can prove that any Galois connection between 2U and 2V arises in this way from a unique relation.

Orthogonal Complement: Let V be a vector space over field K and let V ∗ be the dual space, consisting of linear functions α ∶ V → K. We define the relation ⊥ ⊆ V ∗ × V by

α ⊥ v ⇐⇒ α(v) = 0.

The resulting ⊥⊥-closed subsets are precisely the linear subspaces on both sides. Thus the Fundamental Theorem of Galois Connections gives us an order-reversing bijection between the subspaces of V ∗ and the subspaces of V.

Convex Complement: Let V be a Euclidean space, i.e., a real vector space with an inner product ⟨-,-⟩ ∶ V ×V → ℜ. We define the relation ∼ ⊆ V ×V by

u ∼ v ⇐⇒ ⟨u,v⟩ ≤ 0.

∀ S ⊆ V the operation S ↦ S ∼ ∼ gives the cone genrated by S, thus the ∼ ∼-closed sets are precisely the cones. Here is a picture:

img_20170204_075300

Original Galois Connection: Let L be a field and let G be a finite group of automorphisms of L, i.e., each g ∈ G is a function g ∶ L → L preserving addition and multiplication. We define a relation ∼ ⊆ G × L by

g ∼ l ⇐⇒ g(l) = l.

Define K ∶= L ∼ to be the “subfield fixed by G“. The original Fundamental Theorem of Galois Theory says that the ∼ ∼-closed subsets of G are precisely the subgroups and the ∼ ∼-closed subsets of L are precisely the subfields containing K.

Hilbert’s Nullstellensatz: Let K be a field and consider the ring of polynomials K[x] ∶= K[x1,…,xn] in n commuting variables. For each polynomial f(x) ∶= f(x1,…,xn) ∈ K[x] and for each n-tuple of field elements α ∶= (α1,…,αn) ∈ Kn, we denote the evaluation by f(α) ∶= f(α1,…,αn) ∈ K. Now we define a relation ∼ ⊆ K[x] × Kn by

f(x) ∼ α ⇐⇒ f(α) = 0

By definition, the closure operator ∼ ∼ on subsets of Kn is called the Zariski closure. It is not difficult to prove that it satisfies the additional property of a Kuratowski closure (i.e., finite unions of closed sets are closed) and hence it defines a topology on Kn, called the Zariski topology. Hilbert’s Nullstellensatz says that if K is algebraically closed, then the ∼ ∼-closed subsets of K[x] are precisely the radical ideals (i.e., ideals closed under taking arbitrary roots).

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