Axiom:Rig Axioms

Definition

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

$\struct {S, *, \circ}$ is a rig if and only if it satisifes the axioms:

$(\text A 0)$	$:$	Closure under $*$	$\ds \forall a, b \in S:$	$\ds a * b \in S $
$(\text A 1)$	$:$	Associativity of $*$	$\ds \forall a, b, c \in S:$	$\ds \paren {a * b} * c = a * \paren {b * c} $
$(\text A 2)$	$:$	Commutativity of $*$	$\ds \forall a, b \in S:$	$\ds a * b = b * a $
$(\text A 3)$	$:$	Identity element for $*$: the zero	$\ds \exists 0_S \in S: \forall a \in S:$	$\ds a * 0_S = a = 0_S * a $
$(\text M 0)$	$:$	Closure under $\circ$	$\ds \forall a, b \in S:$	$\ds a \circ b \in S $
$(\text M 1)$	$:$	Associativity of $\circ$	$\ds \forall a, b, c \in S:$	$\ds \paren {a \circ b} \circ c = a \circ \paren {b \circ c} $
$(\text M 2)$	$:$	The zero is a zero element for $\circ$	$\ds \forall a \in S:$	$\ds a \circ 0_S = 0_S = 0_S \circ a $
$(\text D)$	$:$	$\circ$ is distributive over $*$	$\ds \forall a, b, c \in S:$	$\ds a \circ \paren {b * c} = \paren {a \circ b} * \paren {a \circ c}, \paren {a * b} \circ c = \paren {a \circ c} * \paren {a \circ c} $

These criteria are called the rig axioms.

\((\text A 0)\)	$:$	Closure under $*$	\(\ds \forall a, b \in S:\)	\(\ds a * b \in S \)
\((\text A 1)\)	$:$	Associativity of $*$	\(\ds \forall a, b, c \in S:\)	\(\ds \paren {a * b} * c = a * \paren {b * c} \)
\((\text A 2)\)	$:$	Commutativity of $*$	\(\ds \forall a, b \in S:\)	\(\ds a * b = b * a \)
\((\text A 3)\)	$:$	Identity element for $*$: the zero	\(\ds \exists 0_S \in S: \forall a \in S:\)	\(\ds a * 0_S = a = 0_S * a \)
\((\text M 0)\)	$:$	Closure under $\circ$	\(\ds \forall a, b \in S:\)	\(\ds a \circ b \in S \)
\((\text M 1)\)	$:$	Associativity of $\circ$	\(\ds \forall a, b, c \in S:\)	\(\ds \paren {a \circ b} \circ c = a \circ \paren {b \circ c} \)
\((\text M 2)\)	$:$	The zero is a zero element for $\circ$	\(\ds \forall a \in S:\)	\(\ds a \circ 0_S = 0_S = 0_S \circ a \)
\((\text D)\)	$:$	$\circ$ is distributive over $*$	\(\ds \forall a, b, c \in S:\)	\(\ds a \circ \paren {b * c} = \paren {a \circ b} * \paren {a \circ c}, \paren {a * b} \circ c = \paren {a \circ c} * \paren {a \circ c} \)