Orderings on Set with 3 Elements

Examples of Orderings
Let $S = \set {a, b, c}$ be an arbitrary set with $3$ elements.

The following are all the orderings that can be applied to $S$, grouped into isomorphism classes.

In the below, $\tuple {x, y}$ indicates that $x \preccurlyeq y$ for the ordering $\preccurlyeq$ under consideration.

Trivial Ordering
This is its Hasse diagram:
 * Ordering-3-Elements-Trivial.png

$2$ Elements Ordered
This is its Hasse diagram:
 * Ordering-3-Elements-2-Ordered.png

where the labels can be arbitrary.

$2$ Maximal Elements
This is its Hasse diagram:
 * Ordering-3-Elements-2-Maximal.png

where the labels can be arbitrary.

$2$ Minimal Elements
This is its Hasse diagram:
 * Ordering-3-Elements-2-Minimal.png

where the labels can be arbitrary.

Total Orderings
This is its Hasse diagram:
 * Ordering-3-Elements-Total.png

where the labels can be arbitrary.

These orderings are total orderings.

From Totally Ordered Set is Lattice, they are also lattice orderings.

Summary
There are $19$ different orderings that can be applied to $S$, grouped into $5$ isomorphism classes.

Exactly one of those isomorphism classes is of a lattice ordering.

It contains $6$ such orderings.

Hence for a set with $3$ elements, there are $6$ possible lattice orderings that can be applied to that set.