# Definition:Rig

## 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:

\((A0):\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \forall a, b \in S:\) | \(\) | \(\displaystyle \) | \(\) | \(\displaystyle \) | \(\displaystyle a * b \in S\) | \(\displaystyle \) | \(\displaystyle \) | Closure under $*$ | |

\((A1):\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \forall a, b, c \in S:\) | \(\) | \(\displaystyle \) | \(\) | \(\displaystyle \) | \(\displaystyle \left({a * b}\right) * c = a * \left({b * c}\right)\) | \(\displaystyle \) | \(\displaystyle \) | Associativity of $*$ | |

\((A2):\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \forall a, b \in S:\) | \(\) | \(\displaystyle \) | \(\) | \(\displaystyle \) | \(\displaystyle a * b = b * a\) | \(\displaystyle \) | \(\displaystyle \) | Commutativity of $*$ | |

\((A3):\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \exists 0_S \in S: \forall a \in F:\) | \(\) | \(\displaystyle \) | \(\) | \(\displaystyle \) | \(\displaystyle a * 0_S = a = 0_S * a\) | \(\displaystyle \) | \(\displaystyle \) | Identity element for $*$: the zero | |

\((M0):\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \forall a, b \in S:\) | \(\) | \(\displaystyle \) | \(\) | \(\displaystyle \) | \(\displaystyle a \circ b \in S\) | \(\displaystyle \) | \(\displaystyle \) | Closure under $\circ$ | |

\((M1):\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \forall a, b, c \in S:\) | \(\) | \(\displaystyle \) | \(\) | \(\displaystyle \) | \(\displaystyle \left({a \circ b}\right) \circ c = a \circ \left({b \circ c}\right)\) | \(\displaystyle \) | \(\displaystyle \) | Associativity of $\circ$ | |

\((M2):\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \forall a \in S:\) | \(\) | \(\displaystyle \) | \(\) | \(\displaystyle \) | \(\displaystyle a \circ 0_S = 0_S = 0_S \circ a\) | \(\displaystyle \) | \(\displaystyle \) | The zero is a zero element for $\circ$ | |

\((D):\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \forall a, b, c \in S:\) | \(\) | \(\displaystyle \) | \(\) | \(\displaystyle \) | \(\displaystyle a \circ \left({b * c}\right) = \left({a \circ b}\right) * \left({a \circ c}\right), \left({a * b}\right) \circ c = \left({a \circ c}\right) * \left({a \circ c}\right)\) | \(\displaystyle \) | \(\displaystyle \) | $\circ$ 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**:

\((M3):\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \exists 1_S \in S: \forall a \in S:\) | \(\) | \(\displaystyle \) | \(\) | \(\displaystyle \) | \(\displaystyle a \circ 1_S = a = 1_S \circ a\) | \(\displaystyle \) | \(\displaystyle \) | 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 **ri ng** without

*n*egative elements.