Near-semiring

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.

Definition
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:


 * 1) (S, +, 0) is a monoid,
 * 2) (S, ·) is a semigroup,
 * 3) (a + b) · c = a · c + b · c, for all a, b, c in S, and
 * 4) 0 · a = 0 for all a in S.

Near-semirings are a 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 monoid (Г; +, 0), equipped with composition of mappings, pointwise addition of mappings, and the zero function. Subsets of M(Г) closed under the operations provide further examples of near-semirings. Another example is the ordinals under the usual operations of ordinal arithmetic (here Clause 3 should be replaced with its symmetric form c · (a + b) =  c · a + c · b. Strictly speaking, the class of all ordinals is not a set, so the above example should be more appropriately called a class near-semiring. We get a near-semiring in the standard sense if we restrict to those ordinals strictly less than some multiplicatively indecomposable ordinal.