Lecture Slides for MAT-60556 Part VII: Algebraic logic Henri Hansen October 9, 2014 1
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 2
• 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” 3
• Several examples of lattices can be found in math- ematics: 1. (2 X , ⊆ ) 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 4
• 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 5
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 of all formulas of L • Let [ A ] = { B ∈ A | B ≡ A } and B = { [ A ] | A ∈ A} • Let [ A ] ≤ [ B ] mean that A → B is valid 6
• 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 of B (at least B !). The smallest subalgebra A of B such that X ⊆ A is called the algebra generated by X . 7
• 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- order language L , then what would we call a filter? 8
Filters and homomorphisms (II) • Given a boolean algebra B , a set X ⊆ B has the finite meet property iff x 1 ∧ · · · ∧ x n � = 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 | ∃ x 1 , . . . ∃ x n ∈ X ( x 1 ∧ · · · ∧ x n ≤ y ) } is a filter. 9
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 ) ∗ 10
• 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 11
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 12
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 strong if it is closed under finite meets; each filter has a strong base • an atom of B is a minimal non-zero element 13
• 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?)
Recommend
More recommend
