Definitions

# 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 `·', such that (S; +) is a semigroup with identity 0, (S; ·) is a semigroup, these semigroups 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; +, ·) is said to be a near-semiring if it satisfies the following axioms:

1. (S; +) is a semigroup with identity 0,
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

• 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. ISBN 0-7923-5786-8
• 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 Math. 18 (1967), 65-78.
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