# Partial order.
We shall develop the idea of ordinals. To do so we first need the concept of partial ordered sets.
A *partial order* $\le$ on a set $X$ is a binary relation such that $\le$ is
- Reflexive, that $a \le a$ for all $a \in X$,
- Antisymmetric, that $a \le b$ and $b \le a$ implies $a = b$, for all $a,b \in X$, and
- Transitive, that $a \le b$ and $b \le c$ implies $a \le c$, for all $a,b,c \in X$.
In a partial ordered set $(X,\le)$, we need not require $a \le b$ or $b \le a$, in which case $a$ and $b$ are *incomparable*. If however that in a partial ordered set every pair of element is comparable, that $a \le b$ or $b \le a$, then we say it is *totally ordered*, or *linearly ordered*.
For a subset $Y$ of a partial ordered set $X$ with ordering $\le$, we say $b \in Y$ is a *smallest element* of $Y$ if for all $y \in Y$, we have $b \le y$. And we say $m \in Y$ is a *minimal element* of $Y$ if whenever $y \in Y$ is such that $y \le m$, then $y = m$.
Note that if $b \in Y$ is a smallest element of $Y$, then it is unique, and that $b$ is also a minimal element of $Y$. However, in general a minimal element of $Y$ is not necessarily unique, nor is it the smallest.
We can similarly define *largest element* and *maximal element*.