Definition:Rig

Definition

A rig is an additive semiring $\struct {S, *, \circ}$ in which $\struct {S, *}$ is a monoid.

Alternatively, this is a semiring in which $\struct {S, *}$ is a commutative monoid.

That is, $\struct {S, *, \circ}$ is a rig if and only if $\struct {S, *, \circ}$ satisfies 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} $

Note that the zero element needs to be specified here as an axiom: $\text M 2$.

By Ring Product with Zero, in a ring, the property $\text M 2$ of the zero element follows as a consequence of the ring axioms.

Some sources insist on another criterion which a semiring $\struct {S, *, \circ}$ must satisfy to be classified as a rig:

$(\text M 3)$

$:$

Identity element for $\circ$: the unity

$\ds \exists 1_S \in S: \forall a \in S:$

$\ds a \circ 1_S = a = 1_S \circ a $

consistent with the associated definition of a ring as a ring with unity.

Some authors refer to this structure as a semiring.

However, it is the policy of this website to reserve the definition of a semiring for a structure in which $\struct {S, *}$ is only a semigroup.

The term rig originated as a jocular suggestion: a ring without negative elements.

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