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:

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:

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.