# Definition:Semiring (Abstract Algebra)

This page has been identified as a candidate for refactoring of advanced complexity.In particular: Need to distinguish semiring from associative semiringUntil this has been finished, please leave
`{{Refactor}}` in the code.
Because of the underlying complexity of the work needed, it is recommended that you do not embark on a refactoring task until you have become familiar with the structural nature of pages of $\mathsf{Pr} \infty \mathsf{fWiki}$.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Refactor}}` from the code. |

*This page is about Semiring in the context of Abstract Algebra. For other uses, see Semiring.*

## Definition

A **semiring** is a ringoid $\struct {S, *, \circ}$ in which:

That is, such that $\struct {S, *, \circ}$ has the following properties:

\((\text A 0)\) | $:$ | \(\ds \forall a, b \in S:\) | \(\ds a * b \in S \) | ||||||

\((\text A 1)\) | $:$ | \(\ds \forall a, b, c \in S:\) | \(\ds \paren {a * b} * c = a * \paren {b * c} \) | ||||||

\((\text M 0)\) | $:$ | \(\ds \forall a, b \in S:\) | \(\ds a \circ b \in S \) | ||||||

\((\text M 1)\) | $:$ | \(\ds \forall a, b, c \in S:\) | \(\ds \paren {a \circ b} \circ c = a \circ \paren {b \circ c} \) | ||||||

\((\text D)\) | $:$ | \(\ds \forall a, b, c \in S:\) | \(\ds a \circ \paren {b * c} = \paren {a \circ b} * \paren {a \circ c} \) | ||||||

\(\ds \paren {a * b} \circ c = \paren {a \circ c} * \paren {b \circ c} \) |

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 $\struct {S, \circ}$ 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 $\text A 0, \text A 1, \text M 0, \text M 1, \text D$ 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