\begin{definition}[Definition: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:
{{begin-axiom}}
{{axiom | n = \text A 0
        | q = \forall a, b \in S
        | m = a * b \in S
        | rc= Closure under $*$
}}
{{axiom | n = \text A 1
        | q = \forall a, b, c \in S
        | m = \paren {a * b} * c = a * \paren {b * c}
        | rc= Associativity of $*$
}}
{{axiom | n = \text A 2
        | q = \forall a, b \in S
        | m = a * b = b * a
        | rc= Commutativity of $*$
}}
{{axiom | n = \text M 0
        | q = \forall a, b \in S
        | m = a \circ b \in S
        | rc= Closure under $\circ$
}}
{{axiom | n = \text M 1
        | q = \forall a, b, c \in S
        | m = \paren {a \circ b} \circ c = a \circ \paren {b \circ c}
        | rc= Associativity of $\circ$
}}
{{axiom | n = \text D
        | q = \forall a, b, c \in S
        | m = a \circ \paren {b * c} = \paren {a \circ b} * \paren {a \circ c}
        | rc= $\circ$ is distributive over $*$
}}
{{axiom | m = \paren {a * b} \circ c = \paren {a \circ c} * \paren {a \circ c}
        | rc= 
}}
{{end-eqn}}
These criteria are called the '''additive semiring axioms'''.
\end{definition}