nLab
antihomomorphism

Antihomomorphisms

Idea

While a homomorphism of magmas (including groups, rings, etc) must preserve multiplication, an antihomomorphism must instead reverse multiplication.

Definitions

Let AA and BB be magmas, or more generally magma objects in any symmetric monoidal category CC. (Examples include groups, which are magmas with extra properties; rings, which are magma objects in Ab with extra proprties; etc.)

An antihomomorphism from AA to BB is a function (or CC-morphism) f:ABf\colon A \to B such that:

  • for every two (generalised) elements x,yx, y of AA, f(xy)=f(y)f(x)f(x y) = f(y) f(x).

Note that for magma objects in CC, the left-hand side of this equation is a generalised element of BB whose source is |x||y|{|x|} \otimes {|y|} (where |x|{|x|} and |y|{|y|} are the sources of the generalised elements xx and yy and \otimes is the tensor product in CC), while the right-hand side is a generalised element of BB whose source is |y||x|{|y|} \otimes {|x|}. Therefore, this definition only makes unambiguous sense because CC is symmetric monoidal, using the unique natural isomorphism |x||y||y||x|{|x|} \otimes {|y|} \cong {|y|} \otimes {|x|}.

An antiautomorphism is an antihomomorphism whose underlying CC-morphism is an automorphism.

Examples

In a **-algebra the ** operator is an antiautomorphism (in fact an anti-involution).

Created on December 30, 2010 04:54:57 by Toby Bartels (98.19.62.165)