# semiring

A *semiring ^{}* is a set $A$ with two operations

^{}, $+$ and $\cdot $, such that $0\in A$ makes $(A,+)$ into a commutative monoid, $1\in A$ makes $(A,\cdot )$ into a monoid, the operation $\cdot $ distributes (http://planetmath.org/Distributivity) over $+$, and for any $a\in A$, $0\cdot a=a\cdot 0=0$. Usually, $a\cdot b$ is instead written $ab$.

A ring $(R,+,\cdot )$, can be described as a semiring for which $(R,+)$ is
required to be a group. Thus every ring is a semiring.
The natural numbers^{}
$\mathbb{N}$ form a semiring, but not a ring, with the usual multiplication^{} and addition.

Every semiring $A$ has a quasiorder^{} $\u2aaf$
given by $a\u2aafb$ if and only if there exists some $c\in A$ such that $a+c=b$. Any element $a\in A$ with an additive inverse is smaller than
any other element. Thus if $A$ has a nonzero element $a$ with an additive
inverse, then the elements $-a$, $0$, $a$ form a cycle with respect to $\u2aaf$.
If $+$ is an idempotent^{} (http://planetmath.org/Idempotency) operation,
then $\u2aaf$ is a partial order^{}.
Addition and (left and right) multiplication are
order-preserving operators (http://planetmath.org/Poset).

Title | semiring |
---|---|

Canonical name | Semiring |

Date of creation | 2013-03-22 12:27:46 |

Last modified on | 2013-03-22 12:27:46 |

Owner | mps (409) |

Last modified by | mps (409) |

Numerical id | 11 |

Author | mps (409) |

Entry type | Definition |

Classification | msc 16Y60 |

Related topic | Ring |

Related topic | KleeneAlgebra |