An Introduction to Order Theory

Presented by Oliver Scarlet

May 16, 2019 Motivation Relations Orders Bounds and Lattices Links to Other Areas My Research

Why learn about Order Theory?

– Orders can be found in many areas of maths, if you know to look – They can be intuitive algebraic objects and good examples – Order theory has applications in proving difficult theoretical results – It is used in computational type theory and some representation theory – Maths is fun

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Binary Relations

Definition (Binary Relation) A Binary Relation between elements of X and Y is some subset R ⊆ X × Y . If x ∈ X and y ∈ Y are related by R, instead of (x, y) ∈ R we write xRy.

Examples – X = {people}, Y = {houses}, L = {(p, h) | p lives in h}. – X = {people}, Y = {people}, S = {(p, q) | p is shorter than q}. – X = Z, Y = C, R = {(x, y) | xy = 2 or y ≠ 0}. – R = {(x, y) | f (x) = y}.

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Properties of Relations

Definition A relation R ⊆ X × X is – Transitive if xRy and yRz implies xRz. Examples – Reflexive if xRx for all x ∈ X . – R = {(p, q) | p lives with q}. – Anti-Reflexive if xRx is not true for any – R = {(x, y) | x − y ≤ 0} ⊆ Z2. x ∈ X . – R = {(S, T ) | S ⊊ T }. – Symmetric if xRy implies yRx. – v ⊥ u if they are non-zero and – Anti-Symmetric if xRy and yRx implies perpendicular. x = y. – xRy if and only if x is purple. – Connexive if xRy or yRx for all x, y ∈ X .

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Equivalence Relations

Definition (Equivalence Relation) A relation R is an Equivalence Relation if it is – Transitive, – Reflexive, – Symmetric. ∼ Common notation for equivalence relations include =, ≃, ∼, =, ≡.

Examples – xRy if and only if x = y. – x ∼ y iff 1 = 1. – ℓ ∥ m iff ℓ and m are parallel. ∼ – S = T iff there exists a bijection from S to T .

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Equivalence Classes

Definition (Equivalence Class) Given an equivalence relation ∼ over a set X , the Equivalence Class of x is

[x] := {y ∈ X | x ∼ y}.

We write X / ∼ for the set of equivalence classes of X .

Examples – x ∼ y if x = y, then [x] = {x}. – x ∼ y for x, y ∈ X if 1 = 1, then [x] = X . – ℓ ∥ m, then there is one class for each direction. ∼ – S = T when there is a bijection S → T , then there is one class for each cardinality.

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Preorders

Definition (Preorder) A relation R is a Preorder if it is – Transitive, – Reflexive.

Examples – Integers with the relation ≤. – Natural numbers where x|y if there is a ∈ N such that ax = y. – sRt iff to complete subject t you must complete subject s. – P ⇒ Q meaning P implies Q.

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Preorders

Theorem If ≤ is a preorder, then ≥ is a preorder, where x ≥ y if y ≤ x.

Theorem If ≤ is a preorder, then < is a transitive anti-reflexive relation, where x < y if x ≤ y but not y ≤ x.

Definition (Order Preserving) If (X , ≤) and (Y , ⪯) are preordered sets, then a function f : X → Y is Order Preserving if x ≤ y implies f (x) ⪯ f (y).

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Making a Preorder

Theorem For every reflexive relation R ⊆ X 2 there is a preorder ≤ on X where xRy implies x ≤ y.

Proof. We define the preorder as x ≤ y if there is a sequence z1, z2,..., zn ∈ X where xRz1, zn Ry, and each zi Rzi+1. This preorder is called the Transitive Closure of R.

Examples – If we have the relation xRy whenever x is y or a child of y, the transitive closure is x ≤ y when x is a descendent of y. – If we have the relation xRy whenever x is y or a prerequisite subject for y, the transitive closure is x ≤ y if you mast pass x to pass y.

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Special Preorders

Definition (Equivalence Relation) A preorder R is an Equivalence Relation if it is symmetric.

Definition (Partial Order) A preorder R is a Partial Order if it is anti-symmetric. Common notation for partial orders includes ⊆, ≤, ⪯. The set X and some partial order R ⊆ X 2 together form a Poset (X , R).

Examples – (Z, ≤). – (N, |) where x|y if there is a ∈ N such that ax = y. – ({subjects}, ⪯) where s ⪯ t iff to complete t you must complete s.

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Partial Orders

Theorem If ≤ is a partial order, then ≥ is a partial order.

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Hasse Diagrams

How do we draw a poset graphically? For example, {2, 3, 4, 6, 8, 9, 12} ordered by divisibility.

We place x below y when x < y and connect them with a line if there is no z such that x < z < y.

8 12

4 6 9

2 3

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Preorder to Partial Order

Theorem If R is a preorder on X , then x ∼ y iff xRy and yRx defines an equivalence relation on X .

Theorem If R is a preorder on X , and x ∼ y iff xRy and yRx, then [x] ≤ [y] iff xRy defines a partial order on X / ∼.

Examples – S ≤ T if there is an injective function S → T . – x ⪯ y if there is a way to travel from x to y. – x|y for x, y ∈ Z if there is some z ∈ Z with xz = y.

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Total Orders

9

Definition (Total Order) 8 A partial order ≤ is a Total Order if it is connexive. 7 A set equipped with a total order is called a Totally Ordered Set. Examples 6 – (Z, ≤). 5 – Words ordered alphabetically. 4 – [S] ≤ [T ] if there is an injective function S → T (assuming the Axiom of Choice). 3 What does the Hasse Diagram of a Total Order look like? 2 1

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Partial to Total

Definition In a poset X , x is maximal if there is no y ∈ X where x < y.

Theorem Given a finite poset (X , ≤), there is a total ordering ⪯ where x ≤ y implies x ⪯ y. Proof. Let n be the size of X . For each i = n, n − 1,..., 2, 1 pick a maximal element xi ∈ X \{xn ,..., xi+1}. Then let xi ⪯ xj if i ≤ j .

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Partial to Total

x7 = 12 8 12 x6 = 9 x5 = 6 x = 8 4 6 9 4 x3 = 3 x = 4 2 3 2 x1 = 2

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More Orders

Theorem n If ≤ is a preorder on X , we can define a preorder on X where (x1,..., xn ) ≤ (y1,..., yn ) if each xi ≤ yi . We call this the Product Order.

Theorem If ≤ is partial order on X , the product order is a partial order.

Theorem If ≤ is a total order on X , we can define a total order on X n where (x1,..., xn ) ≤ (y1,..., yn ) if there is some i ∈ {1,..., n} where

– xj = yj for j < i and

– xi ≤ yi . We call this the Lexicographical Order.

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Bounds

Definition (Upper and Lower Bound) If ≤ is a preorder on X and S ⊆ X is some subset, then x ∈ X is an Upper Bound of S if s ≤ x for all s ∈ S. x is a Lower Bound of S if it is an upper bound in terms of the preorder ≥.

Definition (Supremum and Infimum) If ≤ is a partial order on X and S ⊆ X is some subset, then x ∈ X is the Supremum of S if it is an upper bound of S, and if whenever y is an upper bound of S, then x ≤ y. We write x = sup S. x is the Infimum of S if it is the supremum in terms of the partial order ≥, and we write this x = inf S.

Theorem In a partial ordering, if S contains its upper bound, then that upper bound is its supremum, and we call it the Maximum of S.

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Bounds

Examples ∪ ∩ ⊆ – For sets ordered by , sup S = s∈S s and inf S = s∈S s. – For natural numbers ordered by divisibility, the supremum is the least common multiple, and the infimum is the greatest common divisor. – In X = {2, 3, 12, 18} ordered by divisibility, S = {2, 3} has upper bounds, but no supremum. – In the set of mathematical statements ordered by implication, sup{P, Q} is P or Q, and inf{P, Q} is P and Q.

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Zorn’s Lemma

Axiom of Choice For any collection of non-empty sets, we can choose one element from each set.

Zorn’s Lemma If X is a partially ordered set such that every subset which is totally ordered has an upper bound, then X has a maximal element.

Theorem The Axiom of Choice is equivalent to Zorn’s Lemma.

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Zorn’s Lemma

Theorem Every set has a total ordering.

Proof. Let X be our set, and define Y to be the set of pairs (S, ≤) where S ⊆ X and ≤ is a total ordering of S. Let us define the partial ordering (S, ≤) ⇒ (T , ⪯) if S ⊆ T and x ≤ y implies x ⪯ y. Then for any totally ordered subset, we can take the union of all the sets and the union of all the partial orders to be an upper bound. Hence, by Zorn’s Lemma, Y has a maximal element. If this element is not (X , ≤), then we can add an element to it that is bigger then everything, contradicting maximality.

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Lattices

Definition (Lattice) A poset X is a Lattice if for every x, y ∈ X , sup{x, y} and inf{x, y} exist.

Why is it called a lattice? What does its Hasse diagram look like?

{A, B, C }

{A, B} {A, C } {B, C }

{A} {B} {C }

∅

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Lattices

Definition (Lattice) A set L equipped with two binary operations ∧, ∨ : L × L → L is called a Lattice if – both operations are commutative, – both operations are associative, – a ∨ (a ∧ b) = a and a ∧ (a ∨ b) = a for all a, b ∈ L.

If we identify a ∧ b = inf{a, b} and a ∨ b = sup{a, b}, and a ≤ b iff a ∧ b = a, we start to see why the two definitions are equivalent.

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Complete Lattices

Definition (Complete Lattice) A lattice is Complete if every set has a supremum and infimum.

Theorem A complete lattice L has a bottom element ⊥ ∈ L and a top element ⊤ ∈ L, where ⊥ ≤ x ≤ ⊤ for all x ∈ L.

Proof. ⊥ = sup ∅ and ⊤ = inf ∅.

Theorem Every finite lattice is complete.

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One Nice Example

The integers ordered by divisibility is a preorder. (x|y iff ∃z s.t. xz = y)

This preorder gives us equivalence classes called associates. (x ∼ y iff x|y and y|x)

Quotienting by this quivalence relation gives as a partial order equivalent to the natural numbers.

Ignoring zero, this partial order is a lattice, but is not complete, since infinite sets have no supremum.

With zero, this partial order is a complete lattice.

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Topology Definition (Directed Set) A preordered set is Directed if every pair of elements has an upper bound in the set. Definition (Upper Set) A subset S of a preordered set is Upper if x ∈ S and x ≤ y implies y ∈ S. Definition (Scott Topology) If X is a poset, a subset S is open in the Scott topology (called Scott-open) if it is upper and for every directed subset D ⊆ X where sup D ∈ S we have S ∩ D ≠ ∅. Theorem Every function which is continuous by the Scott topology (called Scott-continuous) is order preserving. In the case where the sets are finite, all order preserving functions are Scott-continuous. Theorem Consider a topological space (X , τ) and the partial ordering ⊆ on τ. A subset of X is compact if and only if the set of its open neighbourhoods is Scott-open. 26/36 Motivation Relations Orders Bounds and Lattices Links to Other Areas My Research

Category Theory

Definition (Preorder) A category is a preorder if for every pair of objects there is at most one morphism between them.

Definition (Partial Order) A preorder category is a partial order if every isomorphism is the identity.

Definition (Lattice) A partial order category is a lattice if it has binary products and coproducts.

Theorem A partial order category is complete if it has all products and coproducts.

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Category Theory

Theorem If we have a small category, we have a preorder on its objects where X ≤ Y iff there exists some morphism X → Y .

Examples – If we take the category of finite sets with morphisms that are injective, take its preorder, and quotient out by the equivalence relation, we get the natural numbers. – If we take the category of mathematical statements with proofs as morphisms, we get the preorder of implications. – If we take a complete small category, take its preorder, and quotient out by the equivalence relation, we get a complete lattice.

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Computational Type Theory

A type is a label put on values stored by a computer program. They can be used to avoid mistakes in programs.

It is convenient that the input type of functions can be flexible, so that a method that needs a number need not be re-written for every possible way a number can be stored.

To achieve this, there is a preorder <: on the types, where T <: S if a value of type T can be used when a value of type S is expected.

For example, – Integer <: Number. – [first:Integer, second:String] <: [first:Number]. – ∀T .T → X <: ∀T .X → T .

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Type Resolution

How do we figure out the types of variables in a language where they don’t have to be written out?

We can look at how the variable is used to work out what constraints it must satisfy.

Given a set of constraints in some variables, how do we assign a type to each variable that satisfy all the constraints?

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Type Resolution

Theorem (Knaster-Tarski Theorem) If L is a complete lattice and f : L → L is order preserving, then the set of fixed points of f is a complete lattice.

Theorem (Kleene Fixed-Point Theorem) If L is a directed-complete partial order with a top element ⊤, and f : L → L is scott-continuous, then the set of fixed points of f has a top which is

inf{f n (⊤) | n ∈ N}.

Theorem Every finite partial order is directed-complete. So all we need is a partial order of types and a monotone function whose fixed points satisfy all of our constraints.

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Type Resolution

The possible constraints generated are of the following forms:

– ti <: T

– ti <: tj

– ti <: sup{tj , tk } ··· ⇒ – (ti1 <: T1, , tij <: Tj ) (tk <: T )

To resolve each of these we create the monotone function

– (t1,..., tn ) 7→ (t1,..., ti−1, inf{ti , T }, ti+1,..., tn )

– (t1,..., tn ) 7→ (t1,..., ti−1, inf{ti , tj }, ti+1,..., tn )

– (t1,..., tn ) 7→ (t1,..., ti−1, inf{ti , sup{tj , tk }}, ti+1,..., tn }) – etc. whose fixed points are the solutions to the constraints.

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Type Resolution ⊤

⊤number ⊤symbol ⊤record

N1|N2|N3 S1|S2 N1|N2 N2|N3

N1 N2 N3 S1 S2 R1 R2

⊥number ⊥symbol ⊥record

cnumber csymbol nilrecord

⊥

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Type Resolution

So by constructing the type lattice and composing our type solving functions, we can repeatedly apply this to the top object until we reach a fixed point, which will be the most general type assignment that satisfies all the constraints.

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Representations of Symmetric Groups

If we have a ring R and an R-algebra A which has basis C , we can define the relation ←L on C by saying x ←L y if there exists z ∈ C such that the coefficient of x in zy is non-zero. This relation is reflexive.

L Let ≤L be the transitive closure of ←, and let ∼L be the equivalence relation on C where x ∼L y if x ≤L y and y ≤L x, and the equivalence classes of this relation are called left cells. We can lift the preorder ≤L to these cells to get a partial order.

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Representations of Symmetric Groups

For each left cell e, we can define

≤Le C = {x ∈ C | [x] ≤L e}

≤ C

≤ AL(e) = A(C Le )/A(C

In the case where A is the Hecke Algebra of a symmetric group, C is the Kazhdan Lusztig basis, the action of the standard basis on each of these quotient modules gives every irreducible representation of the symmetric group.

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