Semiring Congruence

An Equivalence Relation $\approx\;\subseteq R\times R$ on a Semiring $R$ is a semigroup congruence if

• $x_{1}\approx x_{2}\land y_{1}\approx y_{2}\Rightarrow(x_{1}+y_{1})\approx(x_{2 }+y_{2})$
• $x_{1}\approx x_{2}\land y_{1}\approx y_{2}\Rightarrow(x_{1}\cdot y_{1})\approx (x_{2}\cdot y_{2})$

Properties

• Every Semiring Ideal $I\subseteq R$ induces a congruence $\approx_{I}$ on $R$, defined as

  $\par x\approx_{I}y:=\exists a,b.\;x+a=y+b\par$

  This is known as the Bourne Congruence on $I$.

• Semiring congruences are closed under intersections and unions, so they form a Complete Lattice. Notably, lattices are also semirings, so congruences of semirings form a semiring!