Definition:Rig

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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, $\struct {S, *, \circ}$ has the following properties:


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


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.


Also defined as

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      \(\displaystyle \exists 1_S \in S: \forall a \in S:\) \(\displaystyle a \circ 1_S = a = 1_S \circ a \)             

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


Also known as

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.


Linguistic Note

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