Definition:Rig

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Definition

A rig is an additive semiring $\left({S, *, \circ}\right)$ in which $\left({S, *}\right)$ is a monoid.

Alternatively, this is a semiring in which $\left({S, *}\right)$ is a commutative monoid.


That is, $\left({S, *, \circ}\right)$ has the following properties:


\((A0):\)      \(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(\displaystyle \forall a, b \in S:\) \(\) \(\displaystyle \) \(\displaystyle a * b \in S\) \(\displaystyle \) \(\displaystyle \)          Closure under $*$          
\((A1):\)      \(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(\displaystyle \forall a, b, c \in S:\) \(\) \(\displaystyle \) \(\displaystyle \left({a * b}\right) * c = a * \left({b * c}\right)\) \(\displaystyle \) \(\displaystyle \)          Associativity of $*$          
\((A2):\)      \(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(\displaystyle \forall a, b \in S:\) \(\) \(\displaystyle \) \(\displaystyle a * b = b * a\) \(\displaystyle \) \(\displaystyle \)          Commutativity of $*$          
\((A3):\)      \(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(\displaystyle \exists 0_S \in S: \forall a \in F:\) \(\) \(\displaystyle \) \(\displaystyle a * 0_S = a = 0_S * a\) \(\displaystyle \) \(\displaystyle \)          Identity element for $*$: the zero          
\((M0):\)      \(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(\displaystyle \forall a, b \in S:\) \(\) \(\displaystyle \) \(\displaystyle a \circ b \in S\) \(\displaystyle \) \(\displaystyle \)          Closure under $\circ$          
\((M1):\)      \(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(\displaystyle \forall a, b, c \in S:\) \(\) \(\displaystyle \) \(\displaystyle \left({a \circ b}\right) \circ c = a \circ \left({b \circ c}\right)\) \(\displaystyle \) \(\displaystyle \)          Associativity of $\circ$          
\((M2):\)      \(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(\displaystyle \forall a \in S:\) \(\) \(\displaystyle \) \(\displaystyle a \circ 0_S = 0_S = 0_S \circ a\) \(\displaystyle \) \(\displaystyle \)          The zero is a zero element for $\circ$          
\((D):\)      \(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(\displaystyle \forall a, b, c \in S:\) \(\) \(\displaystyle \) \(\displaystyle a \circ \left({b * c}\right) = \left({a \circ b}\right) * \left({a \circ c}\right), \left({a * b}\right) \circ c = \left({a \circ c}\right) * \left({a \circ c}\right)\) \(\displaystyle \) \(\displaystyle \)          $\circ$ is distributive over $*$          


Note that the zero element needs to be specified here as an axiom: $M2$.

By Ring Product with Zero, in a ring, the property $M2$ of the zero element follows as a consequence of the ring axioms.


Also defined as

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

\((M3):\)      \(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(\displaystyle \exists 1_S \in S: \forall a \in S:\) \(\) \(\displaystyle \) \(\displaystyle a \circ 1_S = a = 1_S \circ a\) \(\displaystyle \) \(\displaystyle \)          Identity element for $\circ$: the unity          

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 $\left({S, *}\right)$ is only a semigroup.


Historical Note

That word rig was originally a jocular suggestion: a ring without negative elements.