Glossary of Order Theory - O

O

• Order-dual. The order dual of a partially ordered set is the same set with the partial order relation replaced by its converse.

• Order-embedding. A function f between posets P and Q is an order-embedding if, for all elements x, y of P, x ≤ y (in P) is equivalent to f(x) ≤ f(y) (in Q).

• Order isomorphism. A mapping f: P → Q between two posets P and Q is called an order isomorphism, if it is bijective and both f and f-1 are monotone. Equivalently, an order isomorphism is a surjective order embedding.

• Order-preserving. See monotone.

• Order-reversing. See antitone.