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Dual Lattice

 

For the lattice (P, ≤ ) the dual is (P, ≥).


The duals are shown in the figure below. The diagram of (P, ≥) can be obtained from that of (P, ≤) by simply turning it upside down.

 

817.png 


Let a * b = meet of a and b = GLB, and a
 b = join of a and b = LUB

Sub-Lattice

 

Let (L, *, ) be a lattice and let S  L. The set (S, *, ) is called sublattice iff it is closed under * and . Sublattice is itself a lattice.
 

Closed Interval.

822.png 

Fig. Interval [a, b]

Let a ≤ b.

Then closed interval of a and b is defined as

[a, b] = [x/ a ≤ x and x ≤ b].


Clearly any closed interval is a chain.

Duality

 The dual of a Boolean expression is obtained by interchanging Boolean sums and Boolean products and interchanging 0’s and 1’s.

e.g., The dual of x  (y  0) is x  (y  1)


Boolean Algebra (Definition)

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Boolean Algebra Homomorphism

 

If (A, +, .,', 0, 1) and (B, , –, 0', 1') are two Boolean algebras, a function h : A  B is called a Boolean algebra homomorphism if h preserves the two binary operation and the unary operations in the following since, for all a, b  A


(a) h(a + b) = h(a) 
 h(b)

(b) h(a  b) = h(a)  h(b)

(c) h (a') = h' (a)


A Boolean homomorphism h : A
 B is a Boolean isomorphism if h is one-to-one onto B.





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