Let $(M,\circ)$ and $(N,\ast)$ be two magmas. I'd like to relax the notion of isomorphism by defining a notion of "weak" isomorphism in the following way: $M$ and $N$ are "weakly" isomorphic if there exist two bijections $\eta,\xi:M\longrightarrow N$ such that \begin{equation} \forall\ x,y\in M,\qquad \eta(x\circ y)=\xi(x)\ast\xi(y) \end{equation} With this definition the magma $(M,\circ)$ defined by taking $M=\{x,y\}$ and setting $x\circ x:=y$, $x\circ y=y\circ x= y\circ y := x$ and the magma $(N,\ast)$ defined by taking $N=\{a,b\}$ and setting $b\circ b:=a$, $a\circ b=b\circ a= a\circ a := b$ are "weakly" isomorphic but not isomorphic.
My question is, is this notion existing and/or used in literature? If yes, could you please point me towards some references?
I think you may be interested in the definition of isotopy (homotopy) of magmas.
Also: Isotope Of A Groupoid on planetmath.org.