Definition:Additive Semiring

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Definition

An additive semiring is a semiring with a commutative distributand.


That is, an additive semiring is a ringoid $\left({S, *, \circ}\right)$ in which:

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


Additive Semiring Axioms

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.


Note on Terminology

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

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


Sources