A name for this property?

In noncommutative ring theory (which might be applied to real matrices), a map $phi:Rto R$ such that $phi(ab)=phi(b)phi(a)$ and $phi(a+b)=phi(a)+phi(b)$ may be called a ring-homomorphism $Rto R^{operatorname{op}}$.

What’s $R^{operatorname{op}}$? Basically if $(R,+,cdot)$ is a ring, $R^{operatorname{op}}$ is the ring $(R,+,cdot^{operatorname{op}})$ given by $acdot^{operatorname{op}}b:=bcdot a$, while the sum remains the same.

I’m not sure if there’s a general name for this (other than “order-reversing homomorphism”, or “a homomorphism $R to R^{op}$”, as said in the answer of G. Sassatelli) but many operations of this nature (including both of the ones you mention) are involutions.

Other examples include:

  • The operation of inversion in any ring: if $a$ and $b$ are invertible elements, then $(ab)^{-1}=b^{-1}a^{-1}$.
  • The operation of transposition on matrices: if $M$ and $N$ are any two matrices for which the product $MN$ is defined, then $(MN)^T = N^T M^T$.

Note that any involution necessarily takes the identity to itself: $M^* = (1 cdot M)^* = M^* cdot 1^* $, hence $1^*=1$.

Edited to add: Just to clarify, not all involutions have the desired property; as Dmitry Rubanovich points out in the comments, an involution does not necessarily reverse order (although most of the interesting ones do). And conversely not every order-reversing homomorphism is an involution; involutions are all bijective and satisfy $x^{**}=x$, which need not be the case for an arbitrary order-reversing homomorphism. But (as I wrote originally) many operations of this nature — including both of the examples in the OP — are involutions, and *-algebras (per Federico Poloni’s comment) provide a rich source of additional examples.

Anti- (automorphism | endomorphism | homomorphism | isomorphism) of (groups | rings | monoids | semigroups | algebras… ).

Sample sentences:

The inversion operation in a group is an antiautomorphism of that group. Matrix transpose is an antiautomorphisms of the ring of matrices. Conjugation is an antiautomorphism of quaternions and octonions. [ More at https://en.wikipedia.org/wiki/Antihomomorphism ]

Those are the most common examples and are also involutions (performing operation twice returns to initial state).

It is more common to say anti*morphism or order-reversing morphism, than to talk about a morphism to the op-structure on the same object.

Leave a Comment