Least Upper Bound and Greatest Lower Bound

In this class, we discuss Least Upper Bound and Greatest Lower Bound.

The reader should have prior knowledge of maximal and minimal elements. Click Here.

Least Upper Bound or Supremum or Join

Take a POSET [S, R]

Let A ⊆ S, x ∈ S we call x as least upper bound if for all y ∈ A It has to satisfy the below conditions.

1) For all y ∈ A yRx should be maintained.

2) We can find multiple x’s. We take the least x.

Example:

POSET P = [S, /]

S = {1, 2, 3, 6, 9, 18}

The diagram below shows the hasse diagram.

LUB(2, 3) = 6

LUB(1, 2) = 2 Point to note 2R2 because POSET.

LUB(2, 9) = 18

Example 2:

The diagram below shows the hasse diagram.

LUB(b, c) = d, e

Two LUB values are not accepted.

Greatest Lower Bound or meet or infimum

Take a POSET [S, R]

Let A ⊆ S, x ∈ S we call x the Greatest Lower bound if for all y ∈ A, It has to satisfy the conditions below.

1) For all y ∈ A xRy should be maintained.

2) We can find multiple x’s. We take the greatest x.

Example:

The diagram below shows the hasse diagram.

GLB(6, 9) = 3

GLB(2, 3) = 1

GLB(1, 2) = 1