# Definition:Semiring (Abstract Algebra)

*This page is about semirings in the context of abstract algebra. For other uses, see Definition:Semiring.*

## Contents

## 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 $\mathsf{Pr} \infty \mathsf{fWiki}$:

- An Additive semiring is a
**semiring**whose distributand is commutative

- A Rig is a
**semiring**whose distributand forms a commutative monoid

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).

## Also see

### Examples

### Stronger properties

- Definition:Commutative Ring with Unity
- Definition:Commutative Ring
- Definition:Ring (Abstract Algebra)
- Definition:Commutative Semiring
- Definition:Rig
- Definition:Additive Semiring