Question

To draw the Hasse diagram of the poset \((S \times S, \le )\).

Short answer

The Hasse diagram of the poset \((S \times S, \le )\) is

Step-by-step solution

Given data

Let \(S = \{ 1,2,3,4\} \) ,with respect to the lexicographic order based on the usual less than elation.

Concept used of lexicographic order

An ordering for the Cartesian product\( \times \)of any two sets\(A\)and\(B\)with order relations\( < A\)and\( < B\), respectively, such that if\(\left( {{a_1},{b_1}} \right)\)and\(\left( {{a_2},{b_2}} \right)\)both belong to\(A \times B\), then\(\left( {{a_1},{b_1}} \right) < \left( {{a_2},{b_2}} \right)\)iff either

1.\({a_1} < A{a_2}\), or

2.\({a_1} = {a_2}\)and\({b_1} < B{b_2}\).

Draw the Hasse diagram

All the pairs of the set \(S \times S\) are as follows:

\(\begin{array}{l}\{ (1,1),(2,1),(3,1),(4,1),(1,2),(2,2),(3,2),(4,2),\\(1,3),(2,3),(3,3),(4,3),(1,4),(2,4),(3,4),(4,4)\} \end{array}\)

Therefore, the Hasse diagram of the poset \((S \times S, \le )\) is