Question

For the given Hasse diagram find the greatest lower bound of \(\{ a,b,c\} \).

Short answer

There is no greatest lower bounds of \(\{ f,g,h\} \).

Step-by-step solution

Given data

Given data is \(\{ a,b,c\} \).

Concept used of greatest lower bound

Let\(A\)be an ordered set and\(X\)a subset of\(A\). An element\(b\)is called a lower bound for the set\(X\)if every element in\(X\)is greater than or equal to\(b\). If such a lower bound exists, the set\(X\)is called bounded below.

Find greatest lower bound

The element \(y\) is called the greatest lower bound of \(A\) if \(y\) is a lower bound of \(A\) and \(z \le y\) whenever \(z\) is a lower bound of \(A\).

Clearly in this Hasse diagram there is no greatest lower bounds of \(\{ f,g,h\} \).