Bounded Unbounded Complemented Lattice

In this class, We discuss Bounded Unbounded Complemented Lattice.

The reader should have prior knowledge of lattice. Click here.

Before we move on to the bounded lattice, we understand the upper and lower bounds.

Upper and Lower Bound:

The Diagram below shows the hasse diagram.

In the Diagram, 18 is called the upper bound because every other element relates to 18.

In the diagram, one is called the lower bound because one is related to all the other elements.

Bounded Lattice:

A lattice is a bounded lattice if the upper and lower bounds exist for the lattice.

Unbounded Lattice:

If upper and lower bounds do not exist for a lattice, then it is an unbounded lattice.

Example:

Lattice L = [I, <=]

I mean a set of all integers.

The Diagram below shows the unbounded lattice.

Complement of an Element:

Upper bound is called “I” and lower bound is called “o”.

If the LUB and GLB of a pair of elements are I and o, then the two elements complement each other.

Consider the below hasse diagram.

LUB(2, 9) = 18 and GLB(2,9) is 1. so complement of 2 is 9.

Complemented lattice:

Consider (L, V, Λ, I, O) as a bounded lattice.

A bounded lattice is said to be a complemented lattice if every element is having a complement.

Example:

L = {(1, 2, 3, 6, 7, 14, 21, 42), /} is a complemented lattice.

The Diagram below shows the hasse Diagram for L.