Axiom:Additive Semiring Axioms

Definition

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.

\((\text A 0)\)	$:$	\(\ds \forall a, b \in S:\)	\(\ds a * b \in S \)	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 $*$
			\(\ds \paren {a * b} \circ c = \paren {a \circ c} * \paren {a \circ c} \)