# Dual Lattice

For the lattice (P, â‰¤ ) the dual is (P, â‰¥).

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

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

# Sub-Lattice

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

**Closed Interval.**

Fig. Interval [a, b]

Let a â‰¤ b.

Then closed interval of a and b is defined as

[a, b] = [x/ a â‰¤ x and x â‰¤ 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 0â€™s and 1â€™s.e.g., The dual of x âˆ§ (y âˆ¨ 0) is x âˆ¨ (y âˆ§ 1)

# Boolean Algebra (Definition)

# Boolean Algebra Homomorphism

If (A, +, .,*'*, 0, 1) and (B, âˆ§, âˆ¨, â€“, 0', 1') are two Boolean algebras, a function h : A â†’ 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 âˆˆ A

(a) h(a + b) = h(a) âˆ¨ h(b)

(b) h(a â‹… b) = h(a) âˆ§ h(b)

(c) h (a*'*) = h*'* (a)

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