Let * (P,≤P)* and

*be posets, and consider two set functions*

**(Q,≤Q)***. We will denote these by*

**∗ ∶ P ⇄ Q ∶ ∗***and*

**p ↦ p ∗***for all*

**q ↦ q ∗***and*

**p ∈ P***. This pair of functions is called a Galois connection if, for all*

**q ∈ Q***and*

**p ∈ P***, we have*

**q ∈ Q****p ≤ P q ∗ ⇐⇒ q ≤ Q p ∗**

Let * ∗ ∶ P ⇄ Q ∶ ∗* be a Galois connection. For all elements

*of*

**x***or*

**P***we will use the notations*

**Q***and*

**x ∗ ∗ ∶= (x ∗)∗***.*

**x ∗ ∗ ∗ ∶= (x ∗ ∗)∗**(1) For all * p ∈ P* and

*we have*

**q ∈ Q*** p ≤ P p ∗ ∗ and q ≤ Q q ∗ ∗*.

(2) For all elements * p1, p2 ∈ P* and

*we have*

**q1, q2 ∈ Q*** p1 ≤ P p2 ⇒ p ∗ 2 ≤ Q p ∗ 1* and

*.*

**q1 ≤ Q q2 ⇒ q2 ∗ ≤ P q1 ∗**(3) For all elements * p ∈ P* and

*we have*

**q ∈ Q*** 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

*we have*

**x***by the reflexivity of partial order. Then from the definition of Galois connection we obtain,*

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

To prove (2) consider elements * x, y* such that

*. From (1) and the transitivity of partial*

**x ≤ y***. Then from the definition of Galois connection we obtain*

**x ≤ y ≤ y ∗ ∗ ⇒ x ≤ y ∗ ∗*** (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

*is a closure operator if it satisfies the following three properties:*

**cl ∶ P → P**(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

*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*

**cl ∶ 2**^{U}→ 2^{U}*.]*

**U**If * ∗ ∶ P → Q ∶ ∗* is a Galois connection, then the basic properties above immediately imply that the compositions

*and*

**∗ ∗ ∶ P → P***are closure operators.*

**∗ ∗ ∶ Q → Q**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

*and*

**∗ ∗ ∶ P → P***. We will say that the element*

**∗ ∗ ∶ Q → Q***is*

**p ∈ P (resp. q ∈ Q)***if*

**∗ ∗-closed***. Then the Galois connection restricts to an order-reversing bijection between the subposets of*

**p∗ ∗ = p (resp. q∗ ∗ = q)***elements.*

**∗ ∗-closed**Proof: Let * Q ∗ ⊆ P* and

*denote the images of the functions*

**P ∗ ⊆ Q***, respectively. The restriction of the connection to these subsets defines an order-reversing bijection:*

**∗ ∶ Q → P and ∗ ∶ P → Q**Indeed, consider any * p ∈ Q ∗*, so that

*for some*

**p = q ∗***. Then by properties (1) and (3) of Galois connections we have*

**q ∈ Q****(p) ∗ ∗ = (q ∗) ∗ ∗ ⇒ p ∗ ∗ = q ∗ ∗ ∗ ⇒ p ∗ ∗ = q ∗ ⇒ p ∗ ∗ = p**

Similarly, for all * q ∈ P ∗* we have

*. The bijections reverse order because of property (2).*

**q ∗ ∗ = q**Finally, note that * Q ∗* and

*are exactly the subsets of*

**P ∗***elements in*

**∗ ∗-closed***and*

**P***, respectively. Indeed, we have seen above that every element of*

**Q***is*

**Q ∗***. Conversely, if*

**∗ ∗-closed***is*

**p ∈ P***then we have*

**∗ ∗-closed*** p = p ∗ ∗ ⇒ p = (p ∗) ∗*,

and it follows that * p ∈ Q ∗*. Similarly, every element of

*is*

**P ∗***.*

**∗ ∗-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

*. Indeed, property (2) tells us that*

**∗ ∗-closed***and then from the universal property of the top element we have*

**1**_{P}≤ 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*

**1**_{P}^{**}= 1_{P}*we conclude that*

**Q***. As a consequence of this, the arbitrary meet of*

**0**^{∗}_{P}= 1_{Q}*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.*

**∗ ∗-closed**Galois connections between Boolean lattices have a particularly nice form, which is closely related to the universal quantifier “* ∀*“. Galois Connections of Boolean Lattices. Let

*be sets and let*

**U,V***be any subset (called a relation) between*

**∼ ⊆ U × V***and*

**U***. As usual, we will write “*

**V***” in place of the statement “*

**u ∼ v***“, and we read this as “*

**(u,v) ∈ ∼***“. Then for all*

**u is related to v***and*

**S ∈ 2**^{U}*we define,*

**T ∈ 2**^{V}* 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

*is a Galois connection,*

**T ↦ T**^{∼}*.*

**∼ ∶ 2**^{U}⇄ 2^{V}∶ ∼To see this, note that ∀ subsets * S ∈ 2^{U}* and

*we have*

**T ∈ 2**^{V}**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

*arises in this way from a unique relation.*

**2**^{V}Orthogonal Complement: Let * V* be a vector space over field

*and let*

**K***be the dual space, consisting of linear functions*

**V ∗***. We define the relation*

**α ∶ V → K***by*

**⊥ ⊆ V ∗ × V*** 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

*and the subspaces of*

**V ∗***.*

**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

*by*

**∼ ⊆ V ×V*** u ∼ v ⇐⇒ ⟨u,v⟩ ≤ 0*.

* ∀ S ⊆ V* the operation

*gives the cone genrated by*

**S ↦ S ∼ ∼***, thus the*

**S***sets are precisely the cones. Here is a picture:*

**∼ ∼-closed**Original Galois Connection: Let * L* be a field and let

*be a finite group of automorphisms of*

**G***, i.e., each*

**L***is a function*

**g ∈ G***preserving addition and multiplication. We define a relation*

**g ∶ L → L***by*

**∼ ⊆ G × L*** g ∼ l ⇐⇒ g(l) = l*.

Define * K ∶= L ∼* to be the “subfield fixed by

*“. The original Fundamental Theorem of Galois Theory says that the*

**G***subsets of*

**∼ ∼-closed***are precisely the subgroups and the*

**G***subsets of*

**∼ ∼-closed***are precisely the subfields containing*

**L***.*

**K**Hilbert’s Nullstellensatz: Let * K* be a field and consider the ring of polynomials

*in n commuting variables. For each polynomial*

**K[x] ∶= K[x**_{1},…,x_{n}]*and for each*

**f(x) ∶= f(x**_{1},…,x_{n}) ∈ K[x]*of field elements*

**n-tuple***, we denote the evaluation by*

**α ∶= (α**_{1},…,α_{n}) ∈ Kn*. Now we define a relation*

**f(α) ∶= f(α**_{1},…,α_{n}) ∈ K*by*

**∼ ⊆ K[x] × Kn****f(x) ∼ α ⇐⇒ f(α) = 0**

By definition, the closure operator * ∼ ∼* on subsets of

*is called the*

**Kn***. It is not difficult to prove that it satisfies the additional property of a*

**Zariski closure***(i.e., finite unions of closed sets are closed) and hence it defines a topology on*

**Kuratowski closure***, called the Zariski topology. Hilbert’s Nullstellensatz says that if*

**Kn***is algebraically closed, then the*

**K***subsets of*

**∼ ∼-closed***are precisely the radical ideals (i.e., ideals closed under taking arbitrary roots).*

**K[x]**