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:


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


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:

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

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.