Is there a special name for matrices A and C if ABC=B I have the following three matrices that satisfy:
$$A\cdot B\cdot C=B$$
Is there a special term to describe the matrices A and C? Or what is the relationship between $A$ and $C$?
I have realized that $A$ and $C$ are in general not inverse matrices of each other, since in general:
$$AB=BC^{-1}\ne C^{-1}B$$
I have also been able to deduce that:
$$B=A^{-1}BC^{-1}$$
by multiplying $A^{-1}$ on the left and $C^{-1}$ on the right on both sides. 
However, this doesn't seem to imply anything special about $A$ and $C$. To make it clear, I am asking if there is any special connection between $A$ and $C$, and/or if there is a special term for it.
Note: For my application, all matrices are $4\times4$ rigid transformation matrices, so they are all invertible, but I am not sure if this is an important requirement or not. 

Assuming that $A,B$ are invertible, then the relation $ABC=B$ implies that $A^{-1}$ and $C$ are similar matrices. Indeed, we have 
$$ C = (AB)^{-1}B = B^{-1}A^{-1}B$$
