A non-empty set with two associative binary operations and , satisfying the distributive laws
In most cases one also assumes that the addition is commutative and that there exists a zero such that for every . The most important examples of semi-rings are rings and distributive lattices (cf. Ring; Distributive lattice). If there is a multiplicative identity 1, the two classes are combined by the condition
The non-negative integers with the usual operations provide an example of a semi-ring that does not satisfy this condition.
Semi-ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Semi-ring&oldid=16685