Sublattices

Sublattices
Definition: Let (L, ≤) be a lattice. A non-empty subset S of L is called a sublattice of L if a Ú b ∈ S and a Ù b ∈ S whenever a ∈ S, b ∈ S.
Or
Let (L, ∨ , ∧ ) be a lattice and let S ⊆ L be a subset of L. Then (S, ∨ , ∧ ) is called a sublattice of (L, ∨ , ∧ ) if and only if S is closed under both operations of join(∨ ) and meet( ∧ ).

From the definition it is clear that sublattice itself is a lattice.
However, any subset of L which is a lattice need not be a sublattice.

For example, consider the lattice shown in the diagram:

We note that
(i) the subset S shown by the diagram below is not a sublattice of L, since a ∧ b ∉ S and a ∨ b ∉ S.

(ii) the set T shown below is not a sublattice of L since a ∨ b ∉ T.

However, T is a lattice when considered as a poset by itself.
(iii) the subset ∪ of L shown below is a sublattice of L:

Example: Let A be any set and P(A) its power set. Then (P(A), ∨ , ∧ ) is a lattice in which join and meet are union of sets and intersection of sets respectively.
A family
of subsets of A such that S ∪ T and S ∩ T are in
for S, T ∈
is a sublattice of (P(A), ∨ , ∧ ). Such a family
is called a ring of subsets of A and is denoted by (R(A), ∨ , ∧ ) (This is not a ring in the sense of algebra). Some author call it lattice of subsets.
Example: The lattice (Dn, ≤ ) is a sublattice of (N, ≤), where ≤ is the relation of divisibility.