Definition:Semiring (Abstract Algebra)

Definition

A semiring or rig is a ringoid $\left({S, *, \circ}\right)$ in which both $\left({S, *}\right)$ and $\left({S, \circ}\right)$ form semigroups.

That is, $\left({S, *, \circ}\right)$ has the following properties:

$(1): \quad \left({S, *, \circ}\right)$ is closed under both $*$ and $\circ$

$(2): \quad$ Both $*$ and $\circ$ are associative on $S$

$(3): \quad \circ$ distributes over $*$

$(4): \quad \exists 0 \in S: \forall a \in S: 0 \circ a = 0 = a \circ 0$: that is, there exists $0$, a zero element of $\left({S, *, \circ}\right)$.

Zero Element

Note that the zero element needs to be specified here as an axiom. In a ring, the existence and nature of such an element follows from the ring axioms.

Note on name

The word rig was originally a jocular suggestion: a ring without negative elements.