Definition:Semiring (Abstract Algebra)

From ProofWiki
Jump to navigation Jump to search

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

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).


Also see

Examples


Stronger properties


Weaker properties