Given a binary relation from to , its opposite relation (or dual, inverse, converse, reverse, etc) is a relation from to as follows:
Note that .
The operation is part of the requirements for Rel to be an allegory.
If is a function thought as a functional entire relation, then is also a function if and only if is a bijection; in that case, is the inverse of .
More generally, we have the following:
If is … | then is … | |
---|---|---|
functional | injective | |
entire | surjective | |
injective | functional | |
surjective | entire |
If is a partial order (or even a preorder), then so is ; so each poset (or proset) has an opposite poset (or proset), which is a special case of an opposite category.
Last revised on May 14, 2022 at 07:48:25. See the history of this page for a list of all contributions to it.