Let R and S be rings and f:RS be a function such that f(r1r2)=f(r2)f(r1) for all r1,r2R.

If f is a homomorphismPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath of the additive groupsMathworldPlanetmath of R and S, then f is called an anti-homomorphsim.

If f is a bijection and anti-homomorphism, then f is called an anti-isomorphism.

If f is an anti-homomorphism and R=S then f is called an anti-endomorphism.

If f is an anti-isomorphism and R=S then f is called an anti-automorphism.

As an example, when mn, the mapping that sends a matrix to its transposeMathworldPlanetmath (or to its conjugate transposeMathworldPlanetmath if the matrix is complex) is an anti-isomorphism of Mm,nMn,m.

R and S are anti-isomorphic if there is an anti-isomorphism RS.

All of the things defined in this entry are also defined for groups.

Title anti-isomorphism
Canonical name Antiisomorphism
Date of creation 2013-03-22 16:01:08
Last modified on 2013-03-22 16:01:08
Owner Mathprof (13753)
Last modified by Mathprof (13753)
Numerical id 15
Author Mathprof (13753)
Entry type Definition
Classification msc 13B10
Classification msc 16B99
Defines anti-endomorphism
Defines anti-homomorphism
Defines anti-isomorphic
Defines anti-automorphism