# 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:

- (
*S*, +, 0) is a monoid, - (
*S*, ·) is a semigroup, - (
*a*+*b*) ·*c*=*a*·*c*+*b*·*c*, for all*a*,*b*,*c*in*S*, and - 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, MR1163371. Kluwer Academic Publishers, Dordrecht, 1999. xii+381 pp. ISBN 0-7923-5786-8 MR1746739 - 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.