# Definition:Dual Ordering

## Definition

Let $\struct {S, \preceq}$ be an ordered set.

Let $\succeq$ be the inverse relation to $\preceq$.

That is, for all $a, b \in S$:

$a \succeq b$ if and only if $b \preceq a$

Then $\succeq$ is called the dual ordering of $\preceq$.

### Dual Ordered Set

The ordered set $\struct {S, \succeq}$ is called the dual ordered set (or just dual) of $\struct {S, \preceq}$.

### Notation for Dual Ordering

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.

### Notation for Dual Strict Ordering

To denote the dual of an strict ordering, the conventional technique is to reverse the symbol.

Thus:

$\succ$ denotes $\prec^{-1}$

and so:

$a \prec b \iff b \succ a$

Similarly for the standard symbol used to denote a strict ordering on numbers:

$>$ denotes $<^{-1}$

and so on.

## Also known as

The dual ordering is also known as the opposite ordering or inverse ordering.

The dual ordering of an ordering $\preccurlyeq$ can also be referred to as the dual of $\preccurlyeq$.

## Also see

• Results about dual orderings can be found here.