antiautomorphism

An antiautomorphism is a bijective map from an algebraic structure to itself that reverses the order of its operation. If S is a structure with a binary operation, an antiautomorphism φ satisfies φ(a · b) = φ(b) · φ(a) for all a, b in S. Equivalently, φ is an isomorphism from S to the opposite structure S^op, where the product is reversed.

Antiautomorphisms occur in several settings. In the algebra of n-by-n matrices over a field, the transpose map T: M_n(F) → M_n(F) is an antiautomorphism since (AB)^T = B^T A^T, and it is involutive (T^2 = identity). In the quaternion algebra H, quaternion conjugation q ↦ q̄ is an antiautomorphism because (q1 q2)̄ = q̄2 q̄1. In group theory, the inversion map g ↦ g^{-1} is an antiautomorphism of any group, as (gh)^{-1} = h^{-1} g^{-1}. In commutative rings, the distinction between automorphisms and antiautomorphisms collapses since ab = ba.

Properties and variants. If φ is a bijective antiautomorphism, the composition φ^2 is typically an automorphism. Many standard antiautomorphisms have order two, making them involutive. The concept is central in the study of structures with symmetry and in the theory of involutions, where an anti-automorphism of order two interacts with additional linear or star-like structures.

See also. Automorphism, opposite ring, opposite group, involution.