Iizuka Congruence

The Iizuka congruence of a Two-Sided Semiring Ideal $I\subseteq R$ is the Semiring Congruence $[\approx]_{I}$ defined as

$\par x[\approx]_{I}y:=\exists r:R.\;\exists a,b\in I.\;x+a+r=y+b+r$

This congruence is a coarser version of the Bourne Congruence that adjusts for the fact that addition in a semiring is in general not Cancellative, and can be thought of a sort of "affine" congruence associated to an ideal.

References

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