Discrete Mathematical Structures by Pearson

Ordered Sets and Lattices 8-17

8.13 SUBLATTICES

Let (L, ∧, ∨) be a lattice. A non-empty subset S of L is called a sub-lattice of L if S is closed under

both operations ∧ and ∨.

That is, if a, b ∈ S ⇒ a ∧ b and a ∨ b ∈ S and thus S by itself is a lattice.

Example 1 Let A={1,2,3} and P(A) ={f,{1},{2},{3},{1, 2},{1, 3},{2, 3}, {1, 2, 3}}.

The subset S

1

={f,{2},{3},{2, 3}} of P(A) is a sub–lattice of (P(A),⊆) while S

2

={f,{2},{3}} of

P(A) is not a sub–lattice of P(A) because sup {2, 3} does not exist.

{1, 3}

{1, 2, 3}

{1

, 2}

{2,

3}

{3}

{1}

{2}

f

Figure 8.20(a)

{3}

{2}

{2, 3}

f

Figure 8.20( b)

{2}

{3}

f

Figure 8.20(c)

Example 2 Let D

n

be the set of all positive divisors of ...