# 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*).^{[1]} If there exists an antiisomorphism between two structures, they are said to be *antiisomorphic.*

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.

## Simple example[edit]

Let *A* be the binary relation (or directed graph) consisting of elements {1,2,3} and binary relation defined as follows:

Let *B* be the binary relation set consisting of elements {*a*,*b*,*c*} and binary relation defined as follows:

Note that the opposite of *B* (denoted *B*^{op}) is the same set of elements with the opposite binary relation (that is, reverse all the arcs of the directed graph):

If we replace *a*, *b*, and *c* with 1, 2, and 3 respectively, we see that each rule in *B*^{op} is the same as some rule in *A*. That is, we can define an isomorphism from *A* to *B*^{op} by . is then an antiisomorphism between *A* and *B*.

## Ring anti-isomorphisms[edit]

Specializing the general language of category theory to the algebraic topic of rings, we have:
Let *R* and *S* be rings and *f*: *R* → *S* be a bijection. Then *f* is a *ring anti-isomorphism*^{[2]} if

If *R* = *S* then *f* is a ring *anti-automorphism*.

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

## Notes[edit]

## References[edit]

- Baer, Reinhold (2005) [1952],
*Linear Algebra and Projective Geometry*, Dover, ISBN 0-486-44565-8 - Jacobson, Nathan (1948),
*The Theory of Rings*, American Mathematical Society, ISBN 0-8218-1502-4 - Pareigis, Bodo (1970),
*Categories and Functors*, Academic Press, ISBN 0-12-545150-4