Opposite relations

Definition

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

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

Note that $\left({R}^{\mathrm{op}}{\right)}^{\mathrm{op}}=R$.

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

Examples

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

More generally, we have the following:

If $R$ is …then ${R}^{\mathrm{op}}$ is …
functionalinjective
entiresurjective
injectivefunctional
surjectiveentire

If $R$ is a partial order (or even a preorder), then so is ${R}^{\mathrm{op}}$; 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)