Prerequisite: Partial Orders and Lattices | Set-1

**Well Ordered Set –**

Given a poset, (X, ≤) we say that ≤ is a well-order (well-ordering) and that is well-ordered by ≤ *iff *every nonempty subset of X has a least element. When X is non-empty, if we pick any two-element subset, {a, b}, of X, since the subset {a, b} must have a least element, we see that either a≤b or b≤a, i.e., every well-order is a total order. **E.g. –** The set of natural number (N) is a well ordered.

**Lattice: **A POSET in which every pair of element has both a least upper bound and greatest lower bound.

### Types of Lattice:-

**1. Bounded Lattice:**

A lattice L is said to be bounded if it has the greatest element I and a least element 0.

E.g. – D_{18}= {1, 2, 3, 6, 9, 18} is a bounded lattice.

**Note:** Every Finite lattice is always bounded.

**2. Complemented Lattice:**

A lattice L is said to be complemented if it is bounded and if every element in L has a complement. Here, each element should have at least one complement.

E.g. – D_{6} {1, 2, 3, 6} is a complemented lattice.

In the above diagram every element has a complement.

**3.Distributive Lattice:**

If a lattice satisfies the following two distribute properties, it is called a distributive lattice.

- x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z)
- x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z)

- A complemented distributive lattice is a boolean algebra or boolean lattice.
- A lattice is distributive if and only if none of its sublattices is isomorphic to N
_{5}or M_{3}. - For distributive lattice each element has unique complement. This can be used as a theorem to prove that a lattice is not distributive.

**4.Modular Lattice**

If a lattice satisfies the following property, it is called a modular lattice.

a^(b∨(a^d)) = (a^b)(a^d).

Example-

Attention reader! Don’t stop learning now. Get hold of all the important CS Theory concepts for SDE interviews with the **CS Theory Course** at a student-friendly price and become industry ready.