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. This dual order P^op is defined to be the set with the inverse order, i.e. x ≤ y holds in P^op iff 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 posets are also said to be duals if they are dually isomorphic, i.e. if one poset is order isomorphic to the dual of the other. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations by or about: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... Order theory is a branch of mathematics that studies various kinds of binary relations that capture the intuitive notion of a mathematical ordering. ... In mathematics, especially order theory, a partially ordered set (or poset for short) is a set equipped with a partial order relation. ... IFF, Iff or iff can stand for: Interchange File Format - a computer file format introduced by Electronic Arts Identification friend or foe - a battlefield identification system iff - the mathematics concept if and only if International Film Festival TIFF - Toronto International Film Festival Montreal International Film Festival This is a disambiguation... In the mathematical area of order theory, a Hasse diagram (pronounced HAHS uh, named after Helmut Hasse (1898–1979)) is a simple picture of a finite partially ordered set. ... In mathematics, a partially ordered set (or poset for short) is a set equipped with a special binary relation which formalizes the intuitive concept of an ordering. ... 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. ...

The importance of this simple definition stems from the fact that each and 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.

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, v) and greatest lower bounds (infima, ^)
• Upper sets and lower sets
• Ideals and filters
• Closure operators and kernel operators

Examples of notions which are self-dual include: In mathematics, the supremum of an ordered set S is the least element (not necessarily in S) which is greater than or equal to each element of S. Consequently, it is also referred to as the least upper bound. ... In mathematics the infimum of a subset of some set is the greatest element, not necessarily in the subset, that is smaller than all other elements of the subset. ... In mathematical order theory, an ideal is a special subset of a partially ordered set. ... In mathematics, a filter is a special subset of a partially ordered set. ... In mathematics, given a partially ordered set (P, ≤), a closure operator on P is a function C : P → P with the following properties: if x ≤ y, then C(x) ≤ C(y), i. ...

• Being a (complete) lattice
• Monotonicity of functions
• Distributivity of lattices, i.e. the lattices for which x ^ (y v z) = (x ^ y) v (x ^ z) holds are exactly those for which the dual statement x v (y ^ z) = (x v y) ^ (x v z) holds
• Being a Boolean algebra
• Being an order isomorphism

A cherry lattice pastry A mathematical lattice is a type of partially ordered set. ... In mathematics, functions between ordered sets are monotonic (or monotone) if they preserve the given order. ... This article may be too technical for most readers to understand. ... 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. ...

See also

• list of Boolean algebra topics