In commutative monoids the finite product is uniquely characterized by recursion on finite subsets, forcing the empty product to equal the neutral element.
Why all rings should have a 1.Mathematics Magazine, 92(1):58–62, 2019
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Finite products in commutative monoids: well-definition, recursion on finite subsets, and why the empty product is $1$
In commutative monoids the finite product is uniquely characterized by recursion on finite subsets, forcing the empty product to equal the neutral element.