Semiring

Dependencies: None

Let $R$ be a set and $+$ and $\circ$ be binary operations. Then $(R, +, \circ)$ is a semiring iff the following statements are true for all $r_1, r_2, r_3 \in R$:

• Additive closure: $r_1 + r_2 \in R$.
• Additive commutativity: $r_1 + r_2 = r_2 + r_1$.
• Additive associativity: $(r_1 + r_2) + r_3 = r_1 + (r_2 + r_3)$.
• Existence of 0: $\exists 0 \in R, \forall r \in R, 0 + r = r + 0 = r$.
• Multiplicative closure: $r_1 \circ r_2 \in R$.
• Mutiplicative associativity: $(r_1 \circ r_2) \circ r_3 = r_1 \circ (r_2 \circ r_3)$.
• Left Distributivity: $r_1 \circ (r_2 + r_3) = (r_1 \circ r_2) + (r_1 \circ r_3)$.
• Right Distributivity: $(r_1 + r_2) \circ r_3 = (r_1 \circ r_3) + (r_2 \circ r_3)$.

Additionally, if $\exists 1 \in R, \forall r \in R, r \circ 1 = 1 \circ r = r$, then $1$ is called a unity of $R$ and $R$ is a semiring with unity.

If $r_1 \circ r_2 = r_2 \circ r_1$, then $R$ is called a commutative semiring.

Dependency for:

1. Identity matrix is identity of matrix product
2. Identity matrix
3. Matrix
4. Square matrices form a (semi)ring
5. Binomial theorem

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• Number of transitive dependencies: 0

Transitive dependencies: None