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).
|Date of creation||2013-03-22 12:27:46|
|Last modified on||2013-03-22 12:27:46|
|Last modified by||mps (409)|