Binary operation on empty set?

Question

Can we techniclly declare a binary operation on an empty set?

Since binary operation does an action on some objects (which empty set dot have)...

Thanks.

Answer 1

Yes, we can. A binary operation on $A$ is a function $A\times A \to A$.

When $A=\emptyset$ we have $\emptyset\times\emptyset=\emptyset$ so a binary operation on $\emptyset$ is a function $\emptyset\to\emptyset$. There is exactly one such function, the empty function.

As for why there is an "empty function" $\emptyset\to\emptyset$ remember that a function $X\to Y$ is a subset of $X\times Y$ satisfying certain axioms. One can see that the empty subset of $\emptyset\times\emptyset$ satisfies these axioms.