Lecture Slides for MAT-60556 Part VII: Algebraic logic Henri Hansen - - PowerPoint PPT Presentation

Lecture Slides for MAT-60556 Part VII: Algebraic logic

Henri Hansen October 9, 2014

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Algebraic approach

• For this part of the course, we think of formulas

being members of a language L; this assumes a fixed alphabet of predicates (finite) and variables (countable).

• For now, we assume the language is of first order,

i.e., that quantification is only allowed for individual variables

• We shall regard logical operators for formulas (∨, ∧, ¬, →)

as algebraic operators

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• The resulting algebra is called Boolean algebra.

Lattices

• A lattice is a nonempty, partially ordered set (L, ≤),

i.e., ≤ is a partial order relation for L: it is transitive, reflexive and antisymmetric.

• Each pair of elements in a lattice has a supremum,

i.e., a minimal element that is larger than both, and an infimum, a maximal element that is smaller than both.

• We denote the supremum of x and y as x ∨ y, and

the infimum as x ∧ y. These are also usually called ”meet” and ”join”

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• Several examples of lattices can be found in math-

ematics:

• 1. (2X, ⊆) for a set X, with ∩ and ∪,
• 2. open sets and set inclusion
• 3. natural numbers with ”is divisor of” (what are

the meet and join?)

• 4. Complex numbers when ≤ is applied to real and

imaginary parts; what are the meet and join?

Properties of lattices

• Given two lattices L and L′, a mapping h : L → L′

is a (lattice) homomorphism iff for all x and y we have

• 1. h(x ∧ y) = h(x) ∧ h(y)
• 2. h(x ∨ y) = h(x) ∨ h(y)
• If a homomorphism is a bijection, it is called an

isomorphism

• If there exists an isomorphism between two lattices,

we say they are isomorphic

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• L′ is a sublattice of L, if L′ ⊆ L and it is closed

under ∧ and ∨

• Given a homomorphism hL → L′, the image of L is

a sublattice of L′ (proof: exercise)

• A lattice is distributive if x∧(y ∨z) = (x∧y)∨(x∧z)

(exercise: prove that this for distributive lattices, the same holds with ∨ and ∧ exchanged!)

• A finite subset of a lattice has both a supremum and

an infimum (proof by induction); Give examples of lattices where this does not hold for infinite subsets.

• If every (infinite) subset of a lattice has an inf and

a sup, the lattice is complete.

• Theorem: If every (infinite) subset has a supremum,

then the lattice is complete!

• 1. Let X be a subset of L and let Y = {z ∈ L | ∀x ∈

X(z ≤ x)}

• 2. Y

has s supremum, and this supremum is the infimum of X!

More on lattices

• If x ≤ y holds for all y, then x is the least element

and if y ≤ x for all y, then y is the greatest element

• A lattice with a least element 0 and greatest ele-

ment 1 is said to be complemented if ∀x∃y(x ∨ y = 1 ∧ x ∧ y = 0). Then we say y is a complement of x.

• Theorem: An element of a distributive lattice has

at most one complement

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Boolean Algebra

• A Boolean algebra is a complemented distributive

lattice with at least two elements. Complement of x is denoted x∗

• Let L be a first-order language, and A be the set
• f all formulas of L
• Let [A] = {B ∈ A | B ≡ A} and B = {[A] | A ∈ A}
• Let [A] ≤ [B] mean that A → B is valid

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• Then (B, ≤) is a boolean algebra with [A] ∧ [B] =

[A ∧ B] and [A]∗ = [¬A]

Properties of Boolean algebras

• A subalgebra of a boolean algebra B is a subset of

B that is closed under boolean operators

• Exercise: Show that the following are equivalent in

a boolean algebra: x ≤ y, y∗ ≤ x∗, x ∧ y∗ = 0, and x∗ ∨ y = 1

• Each subset X of B is included in some subalgebra
• f B (at least B!). The smallest subalgebra A of B

such that X ⊆ A is called the algebra generated by X.

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• The minimal algebra is the 2-element algbera, de-

noted simply 2.

Filters and homomorphisms

• A filter in a boolean algebra B is a nonempty set

F ⊆ B, such that

• 1. If x, y ∈ F then x ∧ y ∈ F
• 2. If x ∈ F and x ≤ y then y ∈ F
• 3. 0 /

∈ F

• If B is the boolean algebra associated with a first-
• rder language L, then what would we call a filter?

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Filters and homomorphisms (II)

• Given a boolean algebra B, a set X ⊆ B has the

finite meet property iff x1 ∧ · · · ∧ xn = 0 for every finite subset of X

• Theorem: A subset X is included in some filter of

B iff it has the finite meet property (fmp) – If X is included in a filter, then clearly property 1 guarantees fmp – If X has the fmp, then the set X+ = {y ∈ B | ∃x1, . . . ∃xn ∈ X(x1 ∧ · · · ∧ xn ≤ y)} is a filter.

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Filters and homomorphisms(III)

• X+ constructed in the theorem is the smallest filter

including X. We say that a filter is principal iff it is generated by a singleton set

• A homomorphism of a boolean algebra B into a

boolean algebra B′ is a mapping h : B → B′ such that for every x, y ∈ B

• 1. h(x ∧ y) = h(x) ∧ h(y)
• 2. h(x ∨ y) = h(x) ∨ h(y)
• 3. h(x∗) = h(x)∗

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• If a homomorphism is a bijection, it is said to be an
• isomorphism. If an isomorphism exists between B

and B′, we say they are isomorphic, denoted B ∼ = B′

• Given a homomorphism h : B → B′ the image h[B]

is a subalgebra of B′.

• Given a homomorphism h, the set h−1(1) = {x ∈

B | h(x) = 1} is a filter in B′, and this filter is called the hull of h

• Given a filter F in B, define the relation x ∼F y to

mean that (x ∧ y) ∨ (x∗ ∧ y∗) ∈ F; it is an equiva- lence and a congruence w.r.t the boolean operators (prove this!)

• The quotient of B w.r.t F, is B/F is the boolean

algebra whose members are x = {y | x ∼F y}

• If F is the hull of the homomorphism h : B → B′

then h[B] is isomorphic to B/F

Ultrafilters

• The filter F is an ultrafilter if it is a maximal filter,

i.e., not strictly included in any other filter; ultrafil- ters have the following (equivalent) properties

• 1. B/F ∼

= 2

• 2. F is the hull of a 2-valued homomorphism on B
• 3. F is an ultrafilter
• 4. Whenever x ∨ y ∈ F then x ∈ F or y ∈ F
• 5. for every x either x ∈ F or x∗ ∈ F

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Ultrafilters (II)

• Theorem: Every filter in a Boolean algebra is in-

cluded in some ultrafilter – The set F of all filters of B is partially ordered by inclusion – Chains of F have upper bounds in F

• Corollaries:
• 1. Any subset of B is included in an ultrafilter if it

is fmp

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• 2. Each element is contained in some ultrafiler
• 3. For any pair of elements there is an ultrafilter

that contains one and not the other

Bases and atoms

• Given a filter F, X ⊆ F is called a base of F if for

each y ∈ F there is some x ∈ X such that x ≤ y

• If X∧ is the set of all finite non-empty meets of any

X ⊆ B, then X∧ generates a filter and is its base.

• A base of F is said to be strongif it is closed under

finite meets; each filter has a strong base

• an atom of Bis a minimal non-zero element

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• Lemma: A non-zero element x is an atom iff the

filter generated by {x} is an ultrafilter

• Theorem: A boolean algebra is finite if and only if

every ultrafilter in B is principal

• A boolean algebra is atomic if for each x = 0, there

is an atom a ∈ B such that a ≤ x

• Examples: Propositional logic is an atomic boolean

algebra, but first-order logic is not (why?)