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, $\struct {S, *, \circ}$ is a rig $\struct {S, *, \circ}$ satisfies the rig axioms:

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:

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.