semiring
A semiring is a set with two operations
, and , such that
makes into a commutative monoid, makes
into a monoid, the operation distributes (http://planetmath.org/Distributivity)
over , and for any , . Usually,
is instead written .
A ring , can be described as a semiring for which is
required to be a group. Thus every ring is a semiring.
The natural numbers
form a semiring, but not a ring, with the usual multiplication
and addition.
Every semiring has a quasiorder
given by if and only if there exists some such that . Any element with an additive inverse is smaller than
any other element. Thus if has a nonzero element with an additive
inverse, then the elements , , form a cycle with respect to .
If is an idempotent
(http://planetmath.org/Idempotency) operation,
then 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 |