# Definition:Additive Semiring/Axioms

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## Definition

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

 $(A0)$ $:$ $\displaystyle \forall a, b \in S:$ $\displaystyle a * b \in S$ Closure under $*$ $(A1)$ $:$ $\displaystyle \forall a, b, c \in S:$ $\displaystyle \left({a * b}\right) * c = a * \left({b * c}\right)$ Associativity of $*$ $(A2)$ $:$ $\displaystyle \forall a, b \in S:$ $\displaystyle a * b = b * a$ Commutativity of $*$ $(M0)$ $:$ $\displaystyle \forall a, b \in S:$ $\displaystyle a \circ b \in S$ Closure under $\circ$ $(M1)$ $:$ $\displaystyle \forall a, b, c \in S:$ $\displaystyle \left({a \circ b}\right) \circ c = a \circ \left({b \circ c}\right)$ Associativity of $\circ$ $(D)$ $:$ $\displaystyle \forall a, b, c \in S:$ $\displaystyle a \circ \left({b * c}\right) = \left({a \circ b}\right) * \left({a \circ c}\right), \left({a * b}\right) \circ c = \left({a \circ c}\right) * \left({a \circ c}\right)$ $\circ$ is distributive over $*$

These criteria are called the additive semiring axioms.