Antiisomorphism

Simple example

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

• 1 \rightarrow 2,
• 1 \rightarrow 3,
• 2 \rightarrow 1.

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

• b \Rightarrow a,
• c \Rightarrow a,
• a \Rightarrow b.

Note that the opposite of B (denoted Bop) is the same set of elements with the opposite binary relation \Leftarrow (that is, reverse all the arcs of the directed graph):

• b \Leftarrow a,
• c \Leftarrow a,
• a \Leftarrow b.

If we replace a, b, and c with 1, 2, and 3 respectively, we see that each rule in Bop is the same as some rule in A. That is, we can define an isomorphism \phi from A to Bop by \phi(1) = a, \phi(2) = b, \phi(3) = c. \phi is then an antiisomorphism between A and B.

Ring anti-isomorphisms

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 if :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: : 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}.

Notes

References

References

1. {{harvnb. Pareigis. 1970
2. {{harvnb. Jacobson. 1948
3. {{harvnb. Baer. 2005