# 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, $\struct {S, *, \circ}$ 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 **ri ng** without

*n*egative elements.