# 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 ring without negative elements.