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An Introduction to Order Theory

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An Introduction to Order Theory 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.
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