Antiisomorphism

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

• ${\displaystyle 1\rightarrow 2,}$
• ${\displaystyle 1\rightarrow 3,}$
• ${\displaystyle 2\rightarrow 1.}$

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

• ${\displaystyle b\Rightarrow a,}$
• ${\displaystyle c\Rightarrow a,}$
• ${\displaystyle a\Rightarrow b.}$

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

• ${\displaystyle b\Leftarrow a,}$
• ${\displaystyle c\Leftarrow a,}$
• ${\displaystyle a\Leftarrow b.}$

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 ${\displaystyle \phi }$ from A to B^op by ${\displaystyle \phi (1)=a,\phi (2)=b,\phi (3)=c}$. ${\displaystyle \phi }$ is then an antiisomorphism between A and B.

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

${\displaystyle f(x+_{R}y)=f(x)+_{S}f(y)\ \ \ {\text{and}}\ \ \ f(x\cdot _{R}y)=f(y)\cdot _{S}f(x)\ \ \ {\text{for all }}x,y\in R.}$

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]

${\displaystyle x_{0}+x_{1}\mathbf {i} +x_{2}\mathbf {j} +x_{3}\mathbf {k} \ \ \mapsto \ \ x_{0}-x_{1}\mathbf {i} -x_{2}\mathbf {j} -x_{3}\mathbf {k} .}$

• 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