In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a near-semiring (also ''seminearring'') is an
algebraic structure
A structure is an arrangement and organization of interrelated elements in a material object or system, or the object or system so organized. Material structures include man-made objects such as buildings and machines and natural objects such as ...
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 operation
In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two.
More specifically, an internal binary op ...
s "+" 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
olan, 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. {{MathSciNet, id=1746739
Krishna, K. V. ''Near-semirings: Theory and application'', Ph.D. thesis, IIT Delhi, New Delhi, India, 2005.
''Near-Rings: The Theory and Its Applications'', Vol. 23 of North-Holland Mathematics Studies, North-Holland Publishing Company, 1983.
* Th
Near Ring Main Pageat the
Johannes Kepler Universität Linz
The Johannes Kepler University Linz (German: ''Johannes Kepler Universität Linz'', short: ''JKU'') is a public institution of higher education in Austria. It is located in Linz, the capital of Upper Austria. It offers bachelor's, master's, d ...
* Willy G. van Hoorn and B. van Rootselaar, ''Fundamental notions in the theory of seminearrings'', Compositio Mathematica v. 18, (1967), pp. 65–78.
Algebraic structures