Let and be magmas, or more generally magma objects in any symmetric monoidal category . (Examples include groups, which are magmas with extra properties; rings, which are magma objects in Ab with extra proprties; etc.)
An antihomomorphism from to is a function (or -morphism) such that:
Note that for magma objects in , the left-hand side of this equation is a generalised element of whose source is (where and are the sources of the generalised elements and and is the tensor product in ), while the right-hand side is a generalised element of whose source is . Therefore, this definition only makes unambiguous sense because is symmetric monoidal, using the unique natural isomorphism .
An antiautomorphism is an antihomomorphism whose underlying -morphism is an automorphism.