Bourne Congruence

The Bourne congruence of a Two-Sided Semiring Ideal I ⊆ R 𝐼 𝑅 I\subseteq R is the Semiring Congruence ≈ I subscript 𝐼 \approx_{I} defined as

x≈Iy:=∃a,b∈I.x+a=y+bformulae-sequencesubscript𝐼𝑥𝑦assign𝑎𝑏𝐼𝑥𝑎𝑦𝑏\par x\approx_{I}y:=\exists a,b\in I.\;x+a=y+b

Clearly, ≈ I subscript 𝐼 \approx_{I} is an Equivalence Relation; the tricky part is showing that it is a congruence with respect to multiplication.

Suppose x 1 ≈ I y 1 subscript 𝐼 subscript 𝑥 1 subscript 𝑦 1 x_{1}\approx_{I}y_{1} and x 2 ≈ I y 2 subscript 𝐼 subscript 𝑥 2 subscript 𝑦 2 x_{2}\approx_{I}y_{2} ; this means that there Merely exists some a 1 , b 1 , a 2 , b 2 ∈ I subscript 𝑎 1 subscript 𝑏 1 subscript 𝑎 2 subscript 𝑏 2 𝐼 a_{1},b_{1},a_{2},b_{2}\in I such that x 1 + a 1 = y 1 + b 1 subscript 𝑥 1 subscript 𝑎 1 subscript 𝑦 1 subscript 𝑏 1 x_{1}+a_{1}=y_{1}+b_{1} and x 2 + a 2 = y 2 + b 2 subscript 𝑥 2 subscript 𝑎 2 subscript 𝑦 2 subscript 𝑏 2 x_{2}+a_{2}=y_{2}+b_{2} . Next, observe that

x1⁢x2+x1⁢a2+a1⁢x2+a1⁢a2subscript𝑥1subscript𝑥2subscript𝑥1subscript𝑎2subscript𝑎1subscript𝑥2subscript𝑎1subscript𝑎2\displaystyle x_{1}x_{2}+x_{1}a_{2}+a_{1}x_{2}+a_{1}a_{2} =(x1+a1)⁢(x2+a2)absentsubscript𝑥1subscript𝑎1subscript𝑥2subscript𝑎2\displaystyle=(x_{1}+a_{1})(x_{2}+a_{2})
=(y1+b1)⁢(y2+b2)absentsubscript𝑦1subscript𝑏1subscript𝑦2subscript𝑏2\displaystyle=(y_{1}+b_{1})(y_{2}+b_{2})
=y1⁢y2+y1⁢b2+b1⁢y2+b1⁢b2absentsubscript𝑦1subscript𝑦2subscript𝑦1subscript𝑏2subscript𝑏1subscript𝑦2subscript𝑏1subscript𝑏2\displaystyle=y_{1}y_{2}+y_{1}b_{2}+b_{1}y_{2}+b_{1}b_{2}

Moreover, both x 1 ⁢ a 2 + a 1 ⁢ x 2 + a 1 ⁢ a 2 subscript 𝑥 1 subscript 𝑎 2 subscript 𝑎 1 subscript 𝑥 2 subscript 𝑎 1 subscript 𝑎 2 x_{1}a_{2}+a_{1}x_{2}+a_{1}a_{2} and y 1 ⁢ b 2 + b 1 ⁢ y 2 + b 1 ⁢ b 2 subscript 𝑦 1 subscript 𝑏 2 subscript 𝑏 1 subscript 𝑦 2 subscript 𝑏 1 subscript 𝑏 2 y_{1}b_{2}+b_{1}y_{2}+b_{1}b_{2} lie within our ideal, so x 1 ⁢ x 2 ≈ I y 1 ⁢ y 2 subscript 𝐼 subscript 𝑥 1 subscript 𝑥 2 subscript 𝑦 1 subscript 𝑦 2 x_{1}x_{2}\approx_{I}y_{1}y_{2} .

Note that this is not the only way we can form a congruence from a Semiring Ideal; see Iizuka Congruence for a slightly coarser version.

References