What is the product of the empty set?

Question

Give:

$fn(S)=\prod_{x\in S}x$

what is:

$fn(\emptyset)$

I can see reason that it would be defined as 1 or 0.

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Note: I thought about restricting the domain of $S$ but that would make the problem less general. If there is no general answer then answers for restricted domains would be valid.

Answer 1

The empty product equals 1.

Answer 2

If you want obvious relation $fn(A\cup B) = fn(A) \cdot fn(B)$ for disjoint $A,B$ to hold, then you don't have any choice, empy product must be multiplicative identity.