Near-semiring

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.

Definition

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:

  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.

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.

Bibliography