Discrete Mathematics (II) Yijia Chen Fudan University Review Partial orders Definition Let A be a set and R ⊆ A2. Then hA; Ri is a partially ordered set (poset) if R is reflexive, antisymmetric and transitive. Definition Let hA; ≤i be a poset. 1. a is maximal if there does not exist b 2 A with a ≤ b and a 6= b. 2. a is minimal if there does not exist b 2 A with b ≤ a and a 6= b. 3. a is maximum if for every b 2 A, we have b ≤ a. 4. a is minimum if for every b 2 A, we have a ≤ b. Definition Let hA; ≤i be a poset and S ⊆ A. 1. u 2 A is an upper bound of S if s ≤ u for every s 2 S. 2. ` 2 A is a lower bound of S if ` ≤ s for every s 2 S. Definition Let hA; ≤i be a poset and S ⊆ A. 1. u is a least upper bound of S, denoted by LUB(S), if u is an upper bound of S and u ≤ u0 for any upper bound u0 of S. 2. ` is a greatest lower bound of S, denoted by GLB(S), if ` is a lower bound of S and `0 ≤ ` for any lower bound `0 of S. Theorem Any S has at most one LUB and at most one GLB. Lattices Definition A lattice is a poset hA; ≤i in which any two elements a; b have an LUB(a; b) and a GLB(a; b). We write a [ b = LUB(a; b) and a \ b = GLB(a; b). We also call them join and meet, respectively. a, b { } a b { } { } ∅ Figure: Hasse diagram of powfa; bg; ⊆ Proposition Any lattice has the following properties: 1. Commutativity: a \ b = b \ a and a [ b = b [ a. 2. Associativity: (a \ b) \ c = a \ (b \ c) and (a [ b) [ c = a [ (b [ c). 3. Idempotent law: a \ a = a and a [ a = a. 4. Absorption law: (a [ b) \ a = a and (a \ b) [ a = a. Algebraic characterization of lattices Definition A semilattice is an algebra S = (S; ∗) satisfying that for all x; y; z 2 S, 1. x ∗ x = x, 2. x ∗ y = y ∗ x, 3. x ∗ (y ∗ z) = (x ∗ y) ∗ z. Proposition Let L = (L; \; [) be a lattice. (A1)( L; [) and (L; \) are two semilattices. (A2)( a [ b) \ a = a and (a \ b) [ a = a for all a; b 2 L. Theorem An algebra L = (L; \; [) satisfying (A1) and (A2) is a lattice. Sublattices and extensions Definition Let L = (L; \; [) be a lattice and S ⊆ L with S 6= ;. Moreover, for all a; b 2 S we have a \ b 2 L and a [ b 2 L: Then S = S; \S ; [S is a sublattice of L. Or we may say that S induces a sublattice of L. Definition If S is a sublattice of L, then L is an extension of S. Ideals Let L = (L; \; [) be a lattice and I ⊆ L. Definition I is an ideal if 1. I induces a sublattice of L (in particular I 6= ;), 2. and a 2 I and b 2 L imply that a \ b 2 I . Definition I is a proper ideal if I is an ideal and I 6= L. Furthermore, if a; b 2 L and a \ b 2 I imply that a 2 I or b 2 I , then I is a prime ideal. An example 1 P a b I 0 Figure: P is prime but I not When is I an ideal? Theorem I is an ideal of L if and only if 1. I 6= ;, 2. I is closed under [, i.e., a; b 2 I implies a [ b 2 I, 3. I is downward closed, i.e., a 2 I and b ≤ a imply b 2 I. Minimal ideals Let H ⊆ L with H 6= ;. What is the minimum ideal I with H ⊆ I ? That is, 1. I is an ideal, 2. if I 0 is an ideal with H ⊆ I 0, then I ⊆ I 0. Let ideal(H) := a 2 L a ≤ a1 [···[ an for some a1;:::; an 2 L with n ≥ 1 : Theorem ideal(H) is the minimum ideal with H ⊆ ideal(H). Principal ideals Definition For every a 2 L, ideal(a) := ideal fag = b ≤ a b 2 L is a principal ideal of L. Theorem ideal(a) = b \ a b 2 L . Some Special Lattices Complete lattices Definition A lattice L is complete if any (finite or infinite) subset A ⊆ L has a greatest lower bound, written \ a; a2A and a greatest lower bound, written [ a: a2A Bounded lattices Definition L is bounded if it has a greatest element (usually denoted by 1) and a least element (usually denoted by 0). Theorem If L is finite, then L is bounded. Complemented lattices Definition A lattice L with 0 and 1 is complemented if for every a 2 L there exists a b such that a [ b = 1 and a \ b = 0. Example For any set S the lattice hpow(S); ⊆i is complemented. Complemented lattices (cont'd) 1 a b c 0 Figure: A complemented lattice where some complements are not unique Distributive lattices Definition A lattice L is distributive if for every a; b; c 2 L 1. a \ (b [ c) = (a \ b) [ (a \ c). 2. a [ (b \ c) = (a [ b) \ (a [ c) Example For any set S the lattice hpow(S); ⊆i is distributive. A non-distributive lattice 1 a b c 0 Figure: a \ (b [ c) 6= (a \ b) [ (a \ c) Boolean Algebras Definition A Boolean algebra is a lattice with 0 and 1 that is distributive and complemented. Example Let A be a set. Then hpow(A); ⊆i is a Boolean algebra. a, b { } a b { } { } Figure: A∅= fa; bg. Example hf1; 2; 3; 6g; ji is a Boolean algebra. Stone's Representation Theorem Theorem (M. H. Stone, 1936) Every finite Boolean algebra is isomorphic to the Boolean algebra hpow(S); ⊆i of a finite set S. Corollary n Every finite Boolean algebra has 2 elements for some n 2 N. Theorem Let B = hA; ≤i be a Boolean algebra. 1. Let a 2 B. Then the complement a0 of a is unique. 2. The mapping a 7! a0 is bijective. 3. Let a; b 2 B. Then (a [ b)0 = a0 \ b0 and (a \ b)0 = a0 [ b0: Proof (1) Let a1; a2 be complements of a. Then a1 = a1 \ 1 = a1 \ (a2 [ a) = (a1 \ a2) [ (a1 \ a) = (a1 \ a2) [ 0 = a1 \ a2: Thus a1 ≤ a2: By symmetry a2 ≤ a1. Thus a1 = a2: Proof (2) By a \ a0 = 0 and a [ a0 = 1 we conclude a0 \ a = 0 and a0 [ a = 1 Then the uniqueness of the complementation implies (a0)0 = a: Thus the mapping a 7! a0 is bijective (why?). Proof (3) (a [ b)0 = a0 \ b0: (a [ b) \ (a0 \ b0) = (a \ a0 \ b0) [ (b \ a0 \ b0) = 0 [ 0 = 0: And (a [ b) [ (a0 \ b0) = (a [ b [ a0) \ (a [ b [ b0) = 1 \ 1 = 1: (a \ b)0 = a0 [ b0: By symmetry. Rings Definition An algebraic structure hR; +; ·i is a ring if 1. hR; +i is an abelian group. 2. hR; ·i is a monoid. 3. Let a; b; c 2 R. Then a · (b + c) = a · b + a · c and (b + c) · a = b · a + c · a. From Boolean algebras to rings Let hB; \; [i be a Boolean algebra. We define for every a; b 2 B: 1. a + b = (a \ b0) [ (a0 \ b) (i.e., the symmetric difference between a and b). 2. a · b = a \ b. Lemma hB; +; ·i is a ring. Boolean rings Definition A Boolean ring hR; +; ·i is a ring with idempotent ·, i.e., a · a = a for every a 2 R. In the definition of the ring derived from a Boolean algebra: a · a = a \ a = a: From Boolean rings to Boolean algebras Let hB; +; ·i be a Boolean ring. We define for every a; b 2 B: 1. a \ b = a · b. 2. a [ b = a + b + ab. Lemma hB; \; [i is a Boolean algebra..