Minimal Maximal Least and Greatest Members

In this class, We discuss Minimal Maximal Least and Greatest Members.

The reader should have prior knowledge of hasse diagram conditions. Click Here.

We take an example and understand the definitions.

Example 1:

Poset P = {(2, 4, 5, 10, 12, 20, 25), /}

The below diagram shows the hasse diagram for the poset P.

Maximal elements are 12, 20, 25.

The minimal elements are 2, 5.

Maximal elements: Let y ∈ P y is maximal if there is no x ∈ P such that yRx

Above 12, we do not have any number, so we call it a maximal number.

Similarly, elements 20 and 25 are maximal.

Minimal members: Let y ∈ P y is called minimal Member if there is no x ∈ P such that xRy.

Below 2, we do not have any number, so we have a minimal number.

Similarly 5.

Example 2:

The diagram below is used to understand the least and greatest Members.

Least Member: y ∈ P, y is called least Member if for all x ∈ P yRx.

Φ is the least Member.

Greatest Member: y ∈ P, y is called greatest Member if for all x ∈ P xRy.

{a, b} is the greatest element.