Partially Ordered Sets

A partially ordered set is a set $S$ with a relation $\le$ on $S$ satisfying:
(1) $a \le a$ for all $a \in S$ (reflexivity)
(2) if $a\le b$ and $b\le a$, then $a=b$ (antisymmetry)
(3) if $a \le b$ and $b \le c$, then $a \le c$ (transitivity)
Note that for two given elements $a$ and $b$, it may not be the case that $a$ and $b$ are comparable, that is, $a \le b$ or $b \le a$. If this is true for all pairs $a$ and $b$ in $S$, we say that $S$ is totally ordered.

The set $S$ of people in the world, with $\le$ defined by "is a direct descendant of," is a partially ordered set. The set $T$ of positive integers, with $\le$ defined by "divides," is a partially ordered set.

Note that neither of these sets are totally ordered. A brother and sister are not comparable in $S$, since they are not direct descendants of each other; and $2$ and $3$ are not comparable in $T$, since neither divides the other.