Euler Characteristic of a Semiring

The Euler characteristic of a Semiring R 𝑅 R , denoted E ⁢ ( R ) 𝐸 𝑅 E(R) is the universal semiring whose additive monoid is Cancellative Monoid.

Explicitly, the universal property of E ⁢ ( R ) 𝐸 𝑅 E(R) is that there is a Semiring Homomorphism ι : R → E ⁢ ( R ) : 𝜄 → 𝑅 𝐸 𝑅 \iota:R\to E(R) , and every other Semiring Homomorphism f : R → S : 𝑓 → 𝑅 𝑆 f:R\to S to a semiring S 𝑆 S with a cancellative additive monoid uniquely factors through ι 𝜄 \iota .

We can explicitly construct E ⁢ ( R ) 𝐸 𝑅 E(R) quotienting R 𝑅 R by the equivalence relation

r∼s:=∃t.r+t=s+tformulae-sequencesimilar-to𝑟𝑠assign𝑡𝑟𝑡𝑠𝑡r\sim s:=\exists t.\;r+t=s+t

.

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