In mathematics, a near-semiring (also seminearring) is an algebraic structure more general to near-ring and semiring. Near-semirings arise naturally from functions on semigroups.
A near-semiring is a nonempty set S with two binary operations `+' and `·', and a constant 0 such that (S; +; 0) is a monoid, (S; ·) is a semigroup, these structures are related by one (right or left) distributive law, and accordingly the 0 is one (right or left, respectively) side absorptive element.
Formally, an algebraic structure (S; +, ·, 0) is said to be a near-semiring if it satisfies the following axioms:
Clearly, near-semirings are common abstraction of semirings and near-rings [Golan, 1999; Pilz, 1983]. The standard examples of near-semirings are typically of the form M(Г), the set of all mappings on a semigroup (Г; +) with identity zero, with respect to pointwise addition and composition of mappings, and certain subsets of this set.