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 ∈ A with a ≤ b and a 6= b. 2. a is minimal if there does not exist b ∈ A with b ≤ a and a 6= b. 3. a is maximum if for every b ∈ A, we have b ≤ a. 4. a is minimum if for every b ∈ A, we have a ≤ b. Definition Let hA, ≤i be a poset and S ⊆ A. 1. u ∈ A is an upper bound of S if s ≤ u for every s ∈ S. 2. ` ∈ A is a lower bound of S if ` ≤ s for every s ∈ 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 pow{a, b}
, ⊆ 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 ∈ 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 ∈ 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 ∈ S we have

a ∩ b ∈ L and a ∪ b ∈ 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 ∈ I and b ∈ L imply that a ∩ b ∈ I .

Definition I is a proper ideal if I is an ideal and I 6= L. Furthermore, if a, b ∈ L and a ∩ b ∈ I imply that a ∈ I or b ∈ 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 ∈ I implies a ∪ b ∈ I, 3. I is downward closed, i.e., a ∈ I and b ≤ a imply b ∈ 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 ∈ L a ≤ a1 ∪ · · · ∪ an for some a1,..., an ∈ L with n ≥ 1 .

Theorem ideal(H) is the minimum ideal with H ⊆ ideal(H). Principal ideals

Definition For every a ∈ L,
 ideal(a) := ideal {a} = b ≤ a b ∈ L is a principal ideal of L.

Theorem  ideal(a) = b ∩ a b ∈ 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; a∈A and a greatest lower bound, written [ a. a∈A 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 ∈ 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 ∈ 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 = {a, b}. Example h{1, 2, 3, 6}, |i 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 ∈ N. Theorem Let B = hA, ≤i be a Boolean algebra.

1. Let a ∈ B. Then the complement a0 of a is unique. 2. The mapping a 7→ a0 is bijective. 3. Let a, b ∈ 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 ∈ 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 ∈ 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 ∈ 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 ∈ B:

1. a ∩ b = a · b. 2. a ∪ b = a + b + ab.

Lemma hB, ∩, ∪i is a Boolean algebra.