Partial Orders and Lattices (Set-2) | Mathematics

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. – D₁₈= {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. – D₆ {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 N₅ or M₃.
• 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-

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