When we can define a binary operation $\cdot:M\times M\rightarrow M$ on an algebraic structure $(M,*)$ such that
If $*$ is associative then $\cdot=*$ even if I'm not sure about the uniqueness (But In right-invertible associative structures this is provable)
If $*$ is right-invetible then $a\cdot b=(a*(b*c))\setminus c$ only if $a\cdot b$ doesn't depends on $c$
So my question is
$1$-There is condition weaker than associativity for $*$ that make us able to define $a\cdot b$?
I am mainly interested in non-associative, right invertible and/or selfdistributive algebraic structures.