Near-semiring

In mathematics, a near-semiring (also "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:
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.