nLab
opposite relation

Opposite relations

Definition

Given a binary relation RR from XX to YY, its opposite relation (or dual, inverse, converse, reverse, etc) is a relation R opR^{op} from YY to XX as follows:

  • bb is R opR^{op}-related to aa if and only if aa is RR-related to bb.

Note that (R op) op=R(R^{op})^{op} = R.

The operation opop is part of the requirements for Rel to be an allegory.

Examples

If ff is a function thought as a functional entire relation, then f opf^{op} is also a function if and only if ff is a bijection; in that case, f opf^{op} is the inverse of ff.

More generally, we have the following:

If RR is …then R opR^{op} is …
functionalinjective
entiresurjective
injectivefunctional
surjectiveentire

If RR is a partial order (or even a preorder), then so is R opR^{op}; so each poset (or proset) has an opposite poset (or proset), which is a special case of an opposite category.

Revised on September 13, 2013 19:46:30 by Toby Bartels (98.23.131.69)