Near-semiring


In mathematics, a near-semiring 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 is a monoid, is a semigroup, these structures are related by a single distributive law, and accordingly 0 is a one-sided absorbing element.
Formally, an algebraic structure is said to be a near-semiring if it satisfies the following axioms:
  1. is a monoid,
  2. is a semigroup,
  3. · 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 . The standard examples of near-semirings are typically of the form M, the set of all mappings on a monoid, equipped with composition of mappings, pointwise addition of mappings, and the zero function. Subsets of M closed under the operations provide further examples of near-semirings. Another example is the ordinals under the usual operations of ordinal arithmetic = 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.