right annihilator

A semiringMathworldPlanetmath is a set A with two operationsMathworldPlanetmath, + and , such that 0A makes (A,+) into a commutative monoid, 1A makes (A,) into a monoid, the operation distributes ( over +, and for any aA, 0a=a0=0. Usually, ab is instead written ab.

A ring (R,+,), can be described as a semiring for which (R,+) is required to be a group. Thus every ring is a semiring. The natural numbersMathworldPlanetmath form a semiring, but not a ring, with the usual multiplicationPlanetmathPlanetmath and addition.

Every semiring A has a quasiorderMathworldPlanetmath given by ab if and only if there exists some cA such that a+c=b. Any element aA 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 . If + is an idempotentPlanetmathPlanetmath ( operation, then is a partial orderMathworldPlanetmath. Addition and (left and right) multiplication are order-preserving operators (

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