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 2/36 Motivation Relations Orders Bounds and Lattices Links to Other Areas My Research Binary Relations Definition (Binary Relation) A Binary Relation between elements of X and Y is some subset R ⊆ X × Y . If x 2 X and y 2 Y are related by R, instead of (x; y) 2 R we write xRy. Examples – X = fpeopleg, Y = fhousesg, L = f(p; h) j p lives in hg. – X = fpeopleg, Y = fpeopleg, S = f(p; q) j p is shorter than qg. – X = Z, Y = C, R = f(x; y) j xy = 2 or y =6 0g. – R = f(x; y) j f (x) = yg. 3/36 Motivation Relations Orders Bounds and Lattices Links to Other Areas My Research Properties of Relations Definition A relation R ⊆ X × X is – Transitive if xRy and yRz implies xRz. Examples – Reflexive if xRx for all x 2 X . – R = f(p; q) j p lives with qg. – Anti-Reflexive if xRx is not true for any – R = f(x; y) j x − y ≤ 0g ⊆ Z2. x 2 X . – R = f(S; T ) j S ( T g. – 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 2 X . 4/36 Motivation Relations Orders Bounds and Lattices Links to Other Areas My Research 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. – ` k m iff ` and m are parallel. ∼ – S = T iff there exists a bijection from S to T . 5/36 Motivation Relations Orders Bounds and Lattices Links to Other Areas My Research Equivalence Classes Definition (Equivalence Class) Given an equivalence relation ∼ over a set X , the Equivalence Class of x is [x] := fy 2 X j x ∼ yg: We write X / ∼ for the set of equivalence classes of X . Examples – x ∼ y if x = y, then [x] = fxg. – x ∼ y for x; y 2 X if 1 = 1, then [x] = X . – ` k 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. 6/36 Motivation Relations Orders Bounds and Lattices Links to Other Areas My Research Preorders Definition (Preorder) A relation R is a Preorder if it is – Transitive, – Reflexive. Examples – Integers with the relation ≤. – Natural numbers where xjy if there is a 2 N such that ax = y. – sRt iff to complete subject t you must complete subject s. – P ) Q meaning P implies Q. 7/36 Motivation Relations Orders Bounds and Lattices Links to Other Areas My Research 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). 8/36 Motivation Relations Orders Bounds and Lattices Links to Other Areas My Research 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 2 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. 9/36 Motivation Relations Orders Bounds and Lattices Links to Other Areas My Research 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; j) where xjy if there is a 2 N such that ax = y. – (fsubjectsg; ⪯) where s ⪯ t iff to complete t you must complete s. 10/36 Motivation Relations Orders Bounds and Lattices Links to Other Areas My Research Partial Orders Theorem If ≤ is a partial order, then ≥ is a partial order. 11/36 Motivation Relations Orders Bounds and Lattices Links to Other Areas My Research Hasse Diagrams How do we draw a poset graphically? For example, f2; 3; 4; 6; 8; 9; 12g 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 12/36 Motivation Relations Orders Bounds and Lattices Links to Other Areas My Research 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. – xjy for x; y 2 Z if there is some z 2 Z with xz = y. 13/36 Motivation Relations Orders Bounds and Lattices Links to Other Areas My Research 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 14/36 Motivation Relations Orders Bounds and Lattices Links to Other Areas My Research Partial to Total Definition In a poset X , x is maximal if there is no y 2 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 2 X n fxn ;:::; xi+1g. Then let xi ⪯ xj if i ≤ j . 15/36 Motivation Relations Orders Bounds and Lattices Links to Other Areas My Research 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 16/36 Motivation Relations Orders Bounds and Lattices Links to Other Areas My Research 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 2 f1;:::; ng where – xj = yj for j < i and – xi ≤ yi . We call this the Lexicographical Order. 17/36 Motivation Relations Orders Bounds and Lattices Links to Other Areas My Research Bounds Definition (Upper and Lower Bound) If ≤ is a preorder on X and S ⊆ X is some subset, then x 2 X is an Upper Bound of S if s ≤ x for all s 2 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 2 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. 18/36 Motivation Relations Orders Bounds and Lattices Links to Other Areas My Research Bounds Examples S T ⊆ – For sets ordered by , sup S = s2S s and inf S = s2S s. – For natural numbers ordered by divisibility, the supremum is the least common multiple, and the infimum is the greatest common divisor. – In X = f2; 3; 12; 18g ordered by divisibility, S = f2; 3g has upper bounds, but no supremum. – In the set of mathematical statements ordered by implication, supfP; Qg is P or Q, and inffP; Qg is P and Q.