Near-semiring
Encyclopedia
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.
, (S; ·) is a semigroup, these structures 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; +, ·, 0) is said to be a near-semiring if it satisfies the following axioms:
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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a near-semiring (also seminearring) is an algebraic
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
structure
Algebraic structure
In abstract algebra, an algebraic structure consists of one or more sets, called underlying sets or carriers or sorts, closed under one or more operations, satisfying some axioms. Abstract algebra is primarily the study of algebraic structures and their properties...
more general to near-ring
Nearring
In mathematics, a near-ring is an algebraic structure similar to a ring, but that satisfies fewer axioms. Near-rings arise naturally from functions on groups.- Definition :...
and semiring
Semiring
In abstract algebra, a semiring is an algebraic structure similar to a ring, but without the requirement that each element must have an additive inverse...
. Near-semirings arise naturally from functions
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
on semigroups.
Definition
A near-semiring is a nonempty set S with two binary operations `+' and `·', and a constant 0 such that (S; +; 0) is a monoidMonoid
In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element. Monoids are studied in semigroup theory as they are naturally semigroups with identity. Monoids occur in several branches of mathematics; for...
, (S; ·) is a semigroup, these structures 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; +, ·, 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.
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.