Antiautomorphisms

An anti-automorphism of an algebraic structure is a bijection from the structure to itself that preserves the underlying set while reversing the order of the operation. In a group G, an anti-automorphism φ satisfies φ(xy) = φ(y) φ(x) for all x, y in G. In a ring R, φ is additive and φ(xy) = φ(y) φ(x). In a field, if the operation is commutative, an anti-automorphism is automatically an automorphism, so the notion is most meaningful in noncommutative settings. More generally, an anti-automorphism is equivalent to an automorphism from the structure to its opposite: φ: A → A is an automorphism of the opposite structure A^op, where multiplication is reversed.

Examples include the transpose map on the matrix ring M_n(F), which is an anti-automorphism since (AB)^T = B^T A^T. The quaternion conjugation x ↦ x̄ is an anti-automorphism, as is the group inverse map x ↦ x^{-1} on any group, since (xy)^{-1} = y^{-1} x^{-1}.

In operator algebras, the star operation in a C*-algebra is a canonical involutive anti-automorphism: (ab)^* = b^* a^*, and (a^*)^* = a; it is typically conjugate-linear as well.

An anti-automorphism may have finite order; if φ^2 is the identity, φ is called involutive. Anti-automorphisms interact with automorphisms and dualities, and they are central in contexts such as representation theory, module duality, and the study of opposite algebras.