# Definition:Semiring (Abstract Algebra)

## Definition

A semiring is a ringoid $\left({S, *, \circ}\right)$ in which:

$(1): \quad \left({S, *}\right)$ forms a semigroup
$(2): \quad \left({S, \circ}\right)$ forms a semigroup.

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

 $(A0)$ $:$ $\displaystyle \forall a, b \in S:$ $\displaystyle a * b \in S$ $(A1)$ $:$ $\displaystyle \forall a, b, c \in S:$ $\displaystyle \left({a * b}\right) * c = a * \left({b * c}\right)$ $(M0)$ $:$ $\displaystyle \forall a, b \in S:$ $\displaystyle a \circ b \in S$ $(M1)$ $:$ $\displaystyle \forall a, b, c \in S:$ $\displaystyle \left({a \circ b}\right) \circ c = a \circ \left({b \circ c}\right)$ $(D)$ $:$ $\displaystyle \forall a, b, c \in S:$ $\displaystyle a \circ \left({b * c}\right) = \left({a \circ b}\right) * \left({a \circ c}\right)$ $\displaystyle \left({a * b}\right) \circ c = \left({a \circ c}\right) * \left({a \circ c}\right)$

These are called the semiring axioms.

## Also defined as

There are various other conventions on what constitutes a semiring.

Some of these have a distinguished, different name on $\mathsf{Pr} \infty \mathsf{fWiki}$:

Still, some sources impose further that there be a identity element for the distributor, that is, that $\left({S, \circ}\right)$ be a monoid.

Such a structure could be referred to as a rig with unity, consistent with the definition of ring with unity.

This website thus specifically defines a semiring as one fulfilling axioms $A0, A1, M0, M1, D$ only (that is, as two semigroups bound by distributivity).