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 bounds (infima, )
- Upper sets and lower sets
- Ideals and filters
- Closure operators and kernel operators.
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 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.
Since partial orders are antisymmetric, the only ones that are self-dual are the equivalence relations.
Read more about this topic: Duality (order Theory)
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