# Near-semiring

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In [mathematics](/source/Mathematics), a **near-semiring**, also called a **seminearring**, is an [algebraic](/source/Abstract_algebra) [structure](/source/Algebraic_structure) more general than a [near-ring](/source/Near-ring) or a [semiring](/source/Semiring). Near-semirings arise naturally from [functions](/source/Function_(mathematics)) on [monoids](/source/Monoid).

## Definition

A near-semiring is a [set](/source/Set_(mathematics)) *S* with two [binary operations](/source/Binary_operation) "+" and "·", and a constant 0 such that (*S*, +, 0) is a monoid (not necessarily [commutative](/source/Commutative_monoid)), (*S*, ·) is a [semigroup](/source/Semigroup), these structures are related by a single (right or left) [distributive law](/source/Distributive_law), and accordingly 0 is a one-sided (right or left, respectively) [absorbing element](/source/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,
1. (*S*, ·) is a semigroup,
1. (*a* + *b*) · *c* = *a* · *c* + *b* · *c*, for all *a*, *b*, *c* in *S*, and
1. 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](/source/Function_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](/source/Ordinal_number) under the usual operations of [ordinal arithmetic](/source/Ordinal_arithmetic) (here Clause 3 should be replaced with its symmetric form *c* · (*a* + *b*) = *c* · *a* + *c* · *b*. Strictly speaking, the [class](/source/Class_(set_theory)) 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](/source/Additively_indecomposable_ordinal#Multiplicatively_indecomposable).

## Bibliography

- [Golan, Jonathan S.](http://math.haifa.ac.il/JSGOLAN/golan2.html), *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. ISBN 0-7923-5786-8
- [Krishna, K. V.](http://www.iitg.ernet.in/kvk), *Near-semirings: Theory and application*, Ph.D. thesis, IIT Delhi, New Delhi, India, 2005.
- [Pilz, G.](http://www.algebra.uni-linz.ac.at/People/gun.html), *Near-Rings: The Theory and Its Applications*, Vol. 23 of North-Holland Mathematics Studies, North-Holland Publishing Company, 1983.
- The [Near Ring Main Page](http://www.algebra.uni-linz.ac.at/Nearrings/) at the [Johannes Kepler Universität Linz](/source/Johannes_Kepler_Universit%C3%A4t_Linz)
- Willy G. van Hoorn and B. van Rootselaar, *Fundamental notions in the theory of seminearrings*, [Compositio Mathematica](/source/Compositio_Mathematica) v. 18, (1967), pp. 65–78.

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