Definition:Semiring (Abstract Algebra)

Definition
A semiring 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:


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


 * Both $*$ and $\circ$ associative on $S$;


 * $\circ$ distributes over $*$.