nLab opposite relation

Definition

Given a binary relation $R$ from $X$ to $Y$, its opposite relation (or dual, inverse, converse, reverse, etc) is a relation $R^{op}$ from $Y$ to $X$ as follows:

• $b$ is $R^{op}$-related to $a$ if and only if $a$ is $R$-related to $b$.

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

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

Examples

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

More generally, we have the following:

If $R$ is …		then $R^{op}$ is …
functional		injective
entire		surjective
injective		functional
surjective		entire

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

See also

• opposite internal relation