## Definition

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.

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.