Marshall Hall, in his famous book *Theory of Groups*, does not always require a binary operation be "well-defined", i.e. an operation is a relation instead of a function (there might be more than one $c$ s.t. $ab=c$). For example, in the discussion of "quasi-group with the inverse property", he use the inverse property to prove that the binary operation is well-defined. This is, I think, different from modern treatment. Could you recommend other references about such treatment?



Hall's treatment of algebraic operations
I am looking at page 7 of Hall's book and I do not see that he is using binary operation in
any non-standard fashion. He defines a "quasigroup with the inverse property" to be a set
with a binary product and unary inverse such that
$$
 a^{-1}(ab) = b = (ba)a^{-1}.
$$
He then wants to show that this is a quasigroup, by which he means a set with a binary operation such that if $ab=c$, then any two of $a$, $b$, $c$ determine the third. This means
he must show that $ax=c$ and $yb=c$ have unique solutions, and that is exactly what he does. That $a$ and $b$ determines $c$ is a consequence of his definition of binary operation.


I can see that Hall's wording is slightly misleading, but he is not using binary operation
in any sense that differs from what we do now. (In his earlier presentation of the axioms for a group he is quite explicit that a product takes a unique value.)


