Lattice Ordering/Examples/Ancestry

Example of Ordering which is not Lattice Ordering
Recall the partial ordering on the set of people:

$D$ is not a lattice ordering.

Proof
Let $a$ and $b$ be siblings.

Let $f$ and $m$ denote the father and mother of both $a$ and $b$.

Then both $f$ and $m$ are an upper bound of $\set {a, b}$.

But it is not necessarily the case that $f$ and $m$ share a common ancestor, unless you grant that either:
 * every two people are somehow descended from the same proto-ancestor, maybe just the first single self-replicating chemical system that may be classified as life

or:
 * both $f$ and $m$ are descended from Adam and Eve, and Eve of course is a descendant of Adam as she came from one of his ribs.

But be that as it may, consider the set:
 * $S = \set {\text {President Buchanan}, \text {President Arthur} }$

Then as $\text {President Buchanan}$ never married and so had no descendants, $S$ has no lower bounds.

Hence $\set {\text {President Buchanan}, \text {President Arthur} }$ has no imfimum.

Hence, by definition, $D$ is not a lattice ordering.