Orderings on Set with 4 Elements

Examples of Orderings

Let $S$ be an arbitrary set with $4$ elements.

The following are the Hasse diagrams for the $15$ isomorphism classes of all the possible orderings on $S$:

Of the above, $\text{(l)}$ and $\text{(o)}$ are lattice orderings.

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 14$: Orderings: Exercise $14.4$