In the

mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

area of

order theory Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article intr ...

, every

partially ordered set In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary ...

''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 In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bic ...

''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 ...

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 ...

s (infima, ∧) * Upper sets and lower sets * Ideals and

filters Filter, filtering or filters may refer to: Science and technology Computing * Filter (higher-order function), in functional programming * Filter (software), a computer program to process a data stream * Filter (video), a software component that ...

Closure 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 de ...

s 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 de ...

s. Examples of notions which are self-dual include: * Being a (

complete Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies t ...

) 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'') holdsThe 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 In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas i ...

* 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).

References

* {{Order theory Order theory

Order theory Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article intr ...

Examples

See also

References