Near-semiring

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.

Bibliography

• Golan, Jonathan S., Semirings and their applications. Updated and expanded version of The theory of semirings, with applications to mathematics and theoretical computer science (Longman Sci. Tech., Harlow, 1992, . Kluwer Academic Publishers, Dordrecht, 1999. xii+381 pp.
• Krishna, K. V., Near-semirings: Theory and application, Ph.D. thesis, IIT Delhi, New Delhi, India, 2005.
• Pilz, G., Near-Rings: The Theory and Its Applications, Vol. 23 of North-Holland Mathematics Studies, North-Holland Publishing Company, 1983.
• The Near Ring Main Page at the Johannes Kepler Universität Linz
• Willy G. van Hoorn and B. van Rootselaar, Fundamental notions in the theory of seminearrings, Compositio Mathematica v. 18, (1967), pp. 65–78.