Axiom:Semiring Axioms

Definition

Let $\struct {S, *, \circ}$ be an algebraic structure.

$\struct {S, *, \circ}$ is a semiring if and only if $\struct {S, *, \circ}$ satisfies the axioms

$(\text A 0)$	$:$	$\ds \forall a, b \in S:$	$\ds a * b \in S $
$(\text A 1)$	$:$	$\ds \forall a, b, c \in S:$	$\ds \paren {a * b} * c = a * \paren {b * c} $
$(\text M 0)$	$:$	$\ds \forall a, b \in S:$	$\ds a \circ b \in S $
$(\text M 1)$	$:$	$\ds \forall a, b, c \in S:$	$\ds \paren {a \circ b} \circ c = a \circ \paren {b \circ c} $
$(\text D)$	$:$	$\ds \forall a, b, c \in S:$	$\ds a \circ \paren {b * c} = \paren {a \circ b} * \paren {a \circ c} $
			$\ds \paren {a * b} \circ c = \paren {a \circ c} * \paren {b \circ c} $

These criteria are called the semiring axioms.

\((\text A 0)\)	$:$	\(\ds \forall a, b \in S:\)	\(\ds a * b \in S \)
\((\text A 1)\)	$:$	\(\ds \forall a, b, c \in S:\)	\(\ds \paren {a * b} * c = a * \paren {b * c} \)
\((\text M 0)\)	$:$	\(\ds \forall a, b \in S:\)	\(\ds a \circ b \in S \)
\((\text M 1)\)	$:$	\(\ds \forall a, b, c \in S:\)	\(\ds \paren {a \circ b} \circ c = a \circ \paren {b \circ c} \)
\((\text D)\)	$:$	\(\ds \forall a, b, c \in S:\)	\(\ds a \circ \paren {b * c} = \paren {a \circ b} * \paren {a \circ c} \)
			\(\ds \paren {a * b} \circ c = \paren {a \circ c} * \paren {b \circ c} \)