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 $14.4$
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