In the
mathematical area of
order theory, every
partially ordered set ''P'' gives rise to a dual (or opposite) partially ordered set which is often denoted by ''P''
op or ''P''
''d''. This dual order ''P''
op is defined to be the same set, but with the inverse order, i.e. ''x'' ≤ ''y'' holds in ''P''
op if and only if ''y'' ≤ ''x'' holds in ''P''. It is easy to see that this construction, which can be depicted by flipping the
Hasse diagram for ''P'' upside down, will indeed yield a partially ordered set. In a broader sense, two partially ordered sets are also said to be duals if they are dually isomorphic, i.e. if one poset is
order isomorphic
In the mathematical field of order theory, an order isomorphism is a special kind of monotone function that constitutes a suitable notion of isomorphism for partially ordered sets (posets). Whenever two posets are order isomorphic, they can be cons ...
to the dual of the other.
The importance of this simple definition stems from the fact that every definition and theorem of order theory can readily be transferred to the dual order. Formally, this is captured by the Duality Principle for ordered sets:
: If a given statement is valid for all partially ordered sets, then its dual statement, obtained by inverting the direction of all order relations and by dualizing all order theoretic definitions involved, is also valid for all partially ordered sets.
If a statement or definition is equivalent to its dual then it is said to be self-dual. Note that the consideration of dual orders is so fundamental that it often occurs implicitly when writing ≥ for the dual order of ≤ without giving any prior definition of this "new" symbol.
Examples
Naturally, there are a great number of examples for concepts that are dual:
*
Greatest elements and least elements
*
Maximal elements and minimal elements
*
Least upper bounds (suprema, ∨) and
greatest lower bound
In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest l ...
s (infima, ∧)
*
Upper sets and lower sets
*
Ideals
Ideal may refer to:
Philosophy
* Ideal (ethics), values that one actively pursues as goals
* Platonic ideal, a philosophical idea of trueness of form, associated with Plato
Mathematics
* Ideal (ring theory), special subsets of a ring considered ...
and
filters
*
Closure operators and
kernel operator In mathematics, a closure operator on a set ''S'' is a function \operatorname: \mathcal(S)\rightarrow \mathcal(S) from the power set of ''S'' to itself that satisfies the following conditions for all sets X,Y\subseteq S
:
Closure operators are d ...
s.
Examples of notions which are self-dual include:
* Being a (
complete)
lattice
*
Monotonicity of functions
*
Distributivity of lattices, i.e. the lattices for which ∀''x'',''y'',''z'': ''x'' ∧ (''y'' ∨ ''z'') = (''x'' ∧ ''y'') ∨ (''x'' ∧ ''z'') holds are exactly those for which the dual statement ∀''x'',''y'',''z'': ''x'' ∨ (''y'' ∧ ''z'') = (''x'' ∨ ''y'') ∧ (''x'' ∨ ''z'') holds
[The quantifiers are essential: for individual elements ''x'', ''y'', ''z'', e.g. the first equation may be violated, but the second may hold; see the N5 lattice for an example.]
* Being a
Boolean algebra
* Being an
order isomorphism.
Since partial orders are
antisymmetric, the only ones that are self-dual are the
equivalence relations (but the notion of partial order is self-dual).
See also
*
Converse relation
*
List of Boolean algebra topics
*
Transpose graph
In the mathematical and algorithmic study of graph theory, the converse, transpose or reverse, entry 2.24 of a directed graph is another directed graph on the same set of vertices with all of the edges reversed compared to the orientation of ...
*
Duality in category theory, of which duality in order theory is a special case
References
*
{{Order theory
Order theory
Order theory