In mathematics, a near-semiring, also called a seminearring, is an algebraic structure more general than a near-ring or a semiring.
Near-semirings arise naturally from functions on monoids.
A near-semiring is a set S with two binary operations "+" and "·", and a constant 0 such that (S, +, 0) is a monoid (not necessarily commutative), (S, ·) is a semigroup, these structures are related by a single (right or left) distributive law, and accordingly 0 is a one-sided (right or left, respectively) absorbing element.
Formally, an algebraic structure (S, +, ·, 0) is said to be a near-semiring if it satisfies the following axioms: Near-semirings are a common abstraction of semirings and near-rings [Golan, 1999; Pilz, 1983].
Subsets of M(Г) closed under the operations provide further examples of near-semirings.