\begin{definition}[Definition:Preordering/Notation]
Symbols used to denote a general preordering relation are usually variants on $\lesssim$, $\precsim$ or $\precapprox$.
A symbol for a preordering can be reversed, and the sense is likewise inverted:
:$a \precsim b \iff b \succsim a$
The notation $a \sim b$ is defined as:
:$a \sim b$ {{iff}} $a \precsim b$ and $b \precsim a$
The notation $a \prec b$ is defined as:
:$a \prec b$ {{iff}} $a \precsim b$ and $a \not \sim b$
\end{definition}