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:

 $(A0):\quad$ $\displaystyle \forall a, b \in S:$  $\displaystyle a * b \in S$ $\quad$ 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.