Definition:Dual Ordering/Notation

Definition

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.

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 14$: Orderings