Antiisomorphism

In category theory, a branch of mathematics, an antiisomorphism (or anti-isomorphism) between structured sets A and B is an isomorphism from A to the opposite of B (or equivalently from the opposite of A to B).

Intuitively, to say that two mathematical structures are antiisomorphic is to say that they are basically opposites of one another.

The concept is particularly useful in an algebraic setting, as, for instance, when applied to rings.

defined as follows: Note that the opposite of B (denoted Bop) is the same set of elements with the opposite binary relation

An example of a ring anti-automorphism is given by the conjugate mapping of quaternions:[3]