\begin{definition}[Definition:Dual Ordering/Notation]
To denote the dual of an ordering, the conventional technique is to reverse the symbol.
Thus:
:$\succeq$ denotes $\preceq^{-1}$
:$\succcurlyeq$ denotes $\preccurlyeq^{-1}$
:$\curlyeqsucc$ denotes $\curlyeqprec^{-1}$
and so:
:$a \preceq b \iff b \succeq a$
:$a \preccurlyeq b \iff b \succcurlyeq a$
:$a \curlyeqprec b \iff b \curlyeqsucc a$
Similarly for the standard symbols used to denote an ordering on numbers:
:$\ge$ denotes $\le^{-1}$
:$\geqslant$ denotes $\leqslant^{-1}$
:$\eqslantgtr$ denotes $\eqslantless^{-1}$
and so on.
\end{definition}