Quotient Semiring

For Rings, there is a nice correspondence between Congruences and Two-Sided Ideals, which is rooted in the fact that the category of rings is a Malcev Category. This means that all Quotient Rings are determined by ideals. Unfortunately, this is a direct consequence of the existence of negation, so for Semirings, we will have to consider quotients by Semiring Congruences.

Explicitly, the quotient semiring $R/\sim$ of a Semiring Congruence $\sim$ is the universal semiring $\iota:R\to R/\sim$ with the property that $x\sim y\Rightarrow\iota(x)=\iota(y)$; EG, it is a Quotient Object.

References

• Golan, Jonathan S. Semirings and Their Applications. Dordrecht: Springer Netherlands, 1999. https://doi.org/10.1007/978-94-015-9333-5.