antimorphs

Antimorphs are mathematical maps that reverse the order of composition between algebraic structures. They generalize the idea of a morphism by reversing the arrangement of operands, while often preserving addition or other basic operations. The term antimorphism is commonly used, and antimorphs frequently appear under the name antihomomorphism in many contexts.

Formally, if A and B are algebraic structures with a binary operation (such as multiplication), a map f: A → B is an antimorphism if for all x, y in A, f(xy) = f(y) f(x). In rings or algebras one also requires f(x + y) = f(x) + f(y). When such a map is bijective, it is called an anti-isomorphism, which identifies A with a “reversed” version of B, known as the opposite structure (for example, the opposite ring uses reversed multiplication).

Common examples include the transpose map on matrices: T: M_n(F) → M_n(F) with T(X) = X^T satisfies (XY)^T = Y^T X^T, making it an antimorphism (and, when bijective, an anti-automorphism of the matrix algebra). In complex matrices, the conjugate transpose X ↦ X̄^T is closely related, combining conjugation with transposition to obtain a conjugate-linear anti-automorphism in certain contexts.

In a broader sense, antimorphs connect to dualities and opposite structures in algebra and category theory. In category-theoretic language, contravariant functors act like antimorphisms by reversing the direction of arrows. See also antimorphism, anti-automorphism, transpose, and opposite category.