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 = 2**^{U} is a Boolean lattice, and if the closure **cl ∶ 2**^{U} → 2^{U} 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:

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.

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 **1**_{P} ≤ P ≤ 1_{p}^{∗∗} and then from the universal property of the top element we have **1**_{P}^{**} = 1_{P}. 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 **0**^{∗}_{P} = 1_{Q}. 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 ∈ 2**^{U} and **T ∈ 2**^{V} we define,

**S**^{∼} ∶= {v ∈ V ∶ ∀ s ∈ S, s ∼ v} ∈ 2^{V},

**T**^{∼} ∶= {u ∈ U ∶ ∀ t ∈ T , u ∼ t} ∈ 2^{U}

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

To see this, note that ∀ subsets **S ∈ 2**^{U} and **T ∈ 2**^{V} 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 **2**^{U} and **2**^{V} 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:

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[x**_{1},…,x_{n}] in n commuting variables. For each polynomial **f(x) ∶= f(x**_{1},…,x_{n}) ∈ K[x] and for each **n-tuple** of field elements **α ∶= (α**_{1},…,α_{n}) ∈ Kn, we denote the evaluation by **f(α) ∶= f(α**_{1},…,α_{n}) ∈ 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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