【离散数学】 SEU - 28 - 2021/06/11 - Lattice and Boolean Algebra
Discrete Mathematical Structures (6th Edition)
2021/06/09 - Lattice and Boolean Algebra
Lattice and Boolean Algebra
Lattice
Homomorphism
Distributive Lattice
Distributive Lattice: < L , ∧ , ∨ > <\!L,∧,∨\!> <L,∧,∨> is a lattice, ∀ a , b , c ∈ L \forall a,b,c∈L ∀a,b,c∈L
a ∧ ( b ∨ c ) = ( a ∧ b ) ∨ ( a ∧ c ) a ∧ ( b ∨ c )=( a ∧ b ) ∨ ( a ∧ c ) a∧(b∨c)=(a∧b)∨(a∧c)
a ∨ ( b ∧ c ) = ( a ∨ b ) ∧ ( a ∨ c ) a ∨ ( b ∧ c )=( a ∨ b ) ∧ ( a ∨ c ) a∨(b∧c)=(a∨b)∧(a∨c)
Note: Actually, distributive lattice only needs to satisfy one of them.
Dimond lattice (not distributive lattice)
Pentagon lattice (not distributive lattice)
Theorem: A lattice L L L is nondistributive if and only if it contains a sublattice that is isomorphic to the diamond lattice or the pentagon lattice.
Greatest (Least) Element
The greatest (least) element a a a：Given lattice < L , ≼ > <\!L, ≼> <L,≼>, for any b b b, we have b ≼ a ( a ≼ b ) b≼a\ (a≼b) b≼a (a≼b)
The greatest (least) element of a lattice is unique, if it exists.
Denoted as 1 ( 0 ) 1 (0) 1(0).
Bounded lattice < L , ≼ > <\!L,≼> <L,≼>: < L , ≼ > <\!L,≼> <L,≼> is a lattice and it has the greatest and least elements.
Denoted as < L , ∧ , ∨ , 1 , 0 > <\! L, ∧, ∨, 1, 0\!> <L,∧,∨,1,0>.
Complement element
Definition: < L , ∧ , ∨ , 0 , 1 > <\! L, ∧, ∨, 0, 1\!> <L,∧,∨,0,1> is a bounded lattice, a ∈ L a∈L a∈L, if there exists b ∈ L b∈L b∈L
a ∧ b = 0 ， a ∨ b = 1 a ∧ b = 0，a ∨ b = 1 a∧b=0，a∨b=1
b b b is the complement element of a a a, denoted as a ′ a' a′.
Definition: < L , ∧ , ∨ , 1 , 0 > <\! L, ∧, ∨, 1, 0\!> <L,∧,∨,1,0> is a bounded lattice, if for any a ∈ L a ∈ L a∈L, a a a has a complement element a ′ a' a′, then L L L is a complemented lattice.
Theorem: < L , ∧ , ∨ , 0 , 1 > \!< L,∧,∨,0,1 \!> <L,∧,∨,0,1> is bounded and distributive. If a ∈ L a∈L a∈L and a a a has its complement element b b b, then b b b is the unique complement element of a a a.
Review
Boolean Algebra
Definition: A Boolean algebra is a lattice that is distributive and complemented.
Example: < P ( A ) , ∪ , ∩ , ∼ , ∅ , A > <\!P(A),∪,∩,\sim,\varnothing,A\!> <P(A),∪,∩,∼,∅,A> is a Boolean algebra.
Theorem: In Boolean Algebra B B B, a , b ∈ B a, b∈B a,b∈B, if a ≼ b a ≼ b a≼b, then we have:
(1) a ∧ b ’ = 0 a ∧ b’= 0 a∧b’=0
(2) a ’ ∨ b = 1 a’∨ b = 1 a’∨b=1
Properties
Theorem: Given < B , ∧ , ∨ , ′ , 0 , 1 > <\!B,∧,∨,',0,1\!> <B,∧,∨,′,0,1>
(1) For every a ∈ B a∈B a∈B, we have ( a ′ ) ′ = a (a')'=a (a′)′=a
(2) For every a , b ∈ B a,b\in B a,b∈B, a and b have complements a ′ , b ′ a',b' a′,b′, then
( a ∧ b ) ′ = a ′ ∨ b ′ , ( a ∨ b ) ′ = a ′ ∧ b ′ (a∧b)'=a'∨b', (a∨b)'=a'∧b' (a∧b)′=a′∨b′,(a∨b)′=a′∧b′
Theorem: < B , ∧ , ∨ , ′ , 0 , 1 > <\!B,∧,∨,',0,1\!> <B,∧,∨,′,0,1> is a Boolean algebra, the following laws hold:
(1) a ∧ b = b ∧ a , a ∨ b = b ∨ a a∧b=b∧a,\ a∨b=b∨a a∧b=b∧a, a∨b=b∨a
(2) ( a ∧ b ) ∧ c = a ∧ ( b ∧ c ) , ( a ∨ b ) ∨ c = a ∨ ( b ∨ c ) (a∧b)∧c=a∧(b∧c),\ (a∨b)∨c=a∨(b∨c) (a∧b)∧c=a∧(b∧c), (a∨b)∨c=a∨(b∨c)
(3) a ∧ a = a , a ∨ a = a a∧a=a,\ a∨a=a a∧a=a, a∨a=a
(4) a ∧ ( a ∨ b ) = a , a ∨ ( a ∧ b ) = a a∧(a∨b)=a,\ a∨(a∧b)=a a∧(a∨b)=a, a∨(a∧b)=a
(5) a ∨ ( b ∧ c ) = ( a ∨ b ) ∧ ( a ∨ c ) , a ∧ ( b ∨ c ) = ( a ∧ b ) ∨ ( a ∧ c ) a∨(b∧c)=(a∨b)∧(a∨c),\ a∧(b∨c)=(a∧b)∨(a∧c) a∨(b∧c)=(a∨b)∧(a∨c), a∧(b∨c)=(a∧b)∨(a∧c)
(6) a ∨ 0 = a , a ∧ 1 = a a∨0=a,\ a∧1=a a∨0=a, a∧1=a
(7) a ∨ 1 = 1 , a ∧ 0 = 0 a∨1=1,\ a∧0=0 a∨1=1, a∧0=0
(8) a ∨ a ′ = 1 , a ∧ a ′ = 0 a∨a'=1,\ a∧a'=0 a∨a′=1, a∧a′=0
(9) a = a ′ ′ a=a'' a=a′′
(10) ( a ∨ b ) ′ = a ′ ∧ b ′ ; ( a ∧ b ) ′ = a ′ ∨ b ′ (a∨b)'=a'∧b';(a∧b)'=a'∨b' (a∨b)′=a′∧b′;(a∧b)′=a′∨b′
Another Definition
< B , ∗ , ⊕ > <\!B, * ,\oplus\!> <B,∗,⊕> is an algebraic system, if ∀ a , b , c ∈ B \forall a,b,c∈B ∀a,b,c∈B the following conditions hold
H1: a ∗ b = b ∗ a , a ⊕ b = b ⊕ a (Commutative Laws) \text{H1: }a*b = b*a, a\oplus b = b\oplus a \tag{Commutative Laws} H1: a∗b=b∗a,a⊕b=b⊕a(Commutative Laws)
H2: a ∗ ( b ⊕ c ) = a ∗ b ⊕ a ∗ c , a ⊕ ( b ∗ c ) = ( a ⊕ b ) ∗ ( a ⊕ c ) (Distributive Laws) \text{H2: } a*(b\oplus c) = a*b\oplus a*c,\ a\oplus(b*c) = (a\oplus b)*(a\oplus c) \tag{Distributive Laws} H2: a∗(b⊕c)=a∗b⊕a∗c, a⊕(b∗c)=(a⊕b)∗(a⊕c)(Distributive Laws)
H3: for 0 and 1 in B , ∀ a ∈ B , a ∗ 1 = a , a ⊕ 0 = a (Identity Laws) \text{H3: for 0 and 1 in }B, \forall a∈B, a*1 = a, a\oplus 0 = a \tag{Identity Laws} H3: for 0 and 1 in B,∀a∈B,a∗1=a,a⊕0=a(Identity Laws)
H 4 : ∀ a ∈ B , there exists a ′ ∈ B , s . t . a ⊕ a ′ = 1 , a ∗ a ′ = 0 (Complementation Laws) H4: \forall a∈B\text{, there exists }a'∈B, s.t.\ a\oplus a' = 1, a*a' = 0 \tag{Complementation Laws} H4:∀a∈B, there exists a′∈B,s.t. a⊕a′=1,a∗a′=0(Complementation Laws)
then < B , ∗ , ⊕ , 0 , 1 > <\!B, * ,\oplus, 0, 1\!> <B,∗,⊕,0,1> is a Boolean algebra.
Sub-Boolean Algebra
Sub-Boolean algebra H H H: < B , ∧ , ∨ , ′ , 0 , 1 > <\!B,∧,∨,',0,1\!> <B,∧,∨,′,0,1> is an algebra
H H H is a subset of B B B,
H H H contains 0 0 0 and 1 1 1,
H H H is closed w.r.t. ∧ , ∨ , ′ ∧, ∨, ' ∧,∨,′
Homomorphism
Homomorphism f f f: < B , ∧ , ∨ , ′ , 0 , 1 > < B,∧,∨,',0,1 > <B,∧,∨,′,0,1> and < B , ⊕ , ⊗ , 0 ‾ , α , β > <\!B,\oplus,\otimes ,\overline{\color{white}0},α,β\!> <B,⊕,⊗,0,α,β> are two Boolean algebra. f f f is a mapping from B B B to B ′ B' B′, satisfying
- f ( a + b ) = f ( a ) ⊕ f ( b ) f(a+b)=f(a)\oplus f(b) f(a+b)=f(a)⊕f(b)
- f ( a ⋅ b ) = f ( a ) ⊗ f ( b ) f(a\cdot b)=f(a)\otimes f(b) f(a⋅b)=f(a)⊗f(b)
- f ( a ′ ) = f ( a ) ‾ f(a')=\overline{f(a)} f(a′)=f(a)
- f ( 0 ) = α , f ( 1 ) = β f(0)=α, f(1)=β f(0)=α,f(1)=β
Atom
a a a covers b b b: b ≤ a b≤a b≤a and b ≠ a b≠a b=a, there is no other element c c c，such that b < c b<c b<c and c < a c<a c<a.
Atom: < B , ∧ , ∨ , ′ , 0 , 1 > <\! B, ∧, ∨, ', 0, 1 \!> <B,∧,∨,′,0,1> is a Boolean algebra，if a ∈ B a∈B a∈B and a a a covers 0 0 0，then a a a is an atom of B B B.
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