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
- Dual Ordering is Ordering demonstrating that a dual ordering is itself indeed also an ordering.
- Results about dual orderings can be found here.
Sources
- 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 14$: Order
- 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.5$: Ordered Sets: Exercise $5$
- 1996: Winfried Just and Martin Weese: Discovering Modern Set Theory. I: The Basics ... (previous) ... (next): Part $1$: Not Entirely Naive Set Theory: Chapter $2$: Partial Order Relations