Definition:Additive Semiring

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An additive semiring is a semiring with a commutative distributand.

That is, an additive semiring is a ringoid $\struct {S, *, \circ}$ in which:

$(1): \quad \struct {S, *}$ forms a commutative semigroup
$(2): \quad \struct {S, \circ}$ forms a semigroup.

Additive Semiring Axioms

An additive semiring is an algebraic structure $\struct {R, *, \circ}$, on which are defined two binary operations $\circ$ and $*$, which satisfy the following conditions:

\((\text A 0)\)   $:$     \(\ds \forall a, b \in S:\) \(\ds a * b \in S \)      Closure under $*$
\((\text A 1)\)   $:$     \(\ds \forall a, b, c \in S:\) \(\ds \paren {a * b} * c = a * \paren {b * c} \)      Associativity of $*$
\((\text A 2)\)   $:$     \(\ds \forall a, b \in S:\) \(\ds a * b = b * a \)      Commutativity of $*$
\((\text M 0)\)   $:$     \(\ds \forall a, b \in S:\) \(\ds a \circ b \in S \)      Closure under $\circ$
\((\text M 1)\)   $:$     \(\ds \forall a, b, c \in S:\) \(\ds \paren {a \circ b} \circ c = a \circ \paren {b \circ c} \)      Associativity of $\circ$
\((\text D)\)   $:$     \(\ds \forall a, b, c \in S:\) \(\ds a \circ \paren {b * c} = \paren {a \circ b} * \paren {a \circ c} \)      $\circ$ is distributive over $*$
\(\ds \paren {a * b} \circ c = \paren {a \circ c} * \paren {a \circ c} \)      

These criteria are called the additive semiring axioms.

Note on Terminology

The term additive semiring was invented by $\mathsf{Pr} \infty \mathsf{fWiki}$ to describe this structure.

As such, it is not generally expected to be seen in this context outside $\mathsf{Pr} \infty \mathsf{fWiki}$.

Most of the literature simply calls this a semiring; however, on $\mathsf{Pr} \infty \mathsf{fWiki}$ the term semiring is reserved for more general structures, not imposing that the distributand be commutative.

Also see