Supremum, Infimum, and Other Orderings

Hasse diagram

The Hasse diagram is used to graphically present a finite ordered set. If $a$ and $b$ are elements of an ordered set, such that $a\preccurlyeq b$, we draw an arrow from $a$ to $b$, as $a\rightarrow b$, and show $b$ placed at a level higher than the level of $a$. We do not draw reflexive and transitive arrows to avoid unnecessary clutter of arrows. This also means that we do not draw self-loops or an arrow from $a$ to $c$ if we have drawn arrows from $a$ to $b$ and $b$ to $c$. If we can reach an element $e_1$ from the element $e_2$ by following the arrows of the Hasse diagram, we call $e_1$and $e_2$comparable. Otherwise, we call them incomparable. Let’s look at an example to appreciate how the Hasse diagram helps us visualize finite ordered sets.

Examples

Let’s consider a set $A = \{1,2,3\}$. The power set ${\cal P} (A)$ is ordered by the subset relation $(\subseteq )$. Let’s see how the Hasse diagram helps us visualize this ordered set.

${\cal P}(A) =\{\emptyset, \{1\}, \{2\}, \{3\}, \{1,2\}, \{1,3\}, \{2,3\}, \{1,2,3\}\}$.