In mathematics

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 semifield is an algebraic 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...

 with two binary operation

Binary operation

In mathematics, a binary operation is a calculation involving two operands, in other words, an operation whose arity is two. Examples include the familiar arithmetic operations of addition, subtraction, multiplication and division....

s, addition and multiplication, which is similar to the field

Field (mathematics)

In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...

, but with some axioms relaxed. There are at least two conflicting conventions of what constitutes a semifield.

• In projective geometry
  Projective geometry
  In mathematics, projective geometry is the study of geometric properties that are invariant under projective transformations. This means that, compared to elementary geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts...
  
   and finite geometry
  Finite geometry
  A finite geometry is any geometric system that has only a finite number of points.Euclidean geometry, for example, is not finite, because a Euclidean line contains infinitely many points, in fact as many points as there are real numbers...
  
   (MSC
  Mathematics Subject Classification
  The Mathematics Subject Classification is an alphanumerical classification scheme collaboratively produced by staff of and based on the coverage of the two major mathematical reviewing databases, Mathematical Reviews and Zentralblatt MATH...
  
   51A, 51E, 12K10), a semifield is a ring (S,+,·) where (S,+) is an abelian group
  Abelian group
  In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
  
   with identity element
  Identity element
  In mathematics, an identity element is a special type of element of a set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them...
  
   0, the multiplication is distributive
  Distributivity
  In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalizes the distributive law from elementary algebra.For example:...
  
   with respect to the addition on the left and on the right, and (S,·) is a division ring
  Division ring
  In abstract algebra, a division ring, also called a skew field, is a ring in which division is possible. Specifically, it is a non-trivial ring in which every non-zero element a has a multiplicative inverse, i.e., an element x with...
  
    that is not assumed to be commutative or associative. This structure is a special case of a quasifield
  Quasifield
  In mathematics, a quasifield is an algebraic structure where + and \cdot are binary operations on Q, much like a division ring, but with some weaker conditions.-Definition:A quasifield is a structure, where + and...
  
  . If S is a finite set, being a division ring is equivalent to the absence of zero divisors, so that a·b = 0 implies that a = 0 or b = 0.

• In ring theory
  Ring theory
  In abstract algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers...
  
  , combinatorics
  Combinatorics
  Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...
  
  , functional analysis
  Functional analysis
  Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
  
  , and theoretical computer science
  Theoretical computer science
  Theoretical computer science is a division or subset of general computer science and mathematics which focuses on more abstract or mathematical aspects of computing....
  
  , a semifield is a 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...
  
   (MSC
  Mathematics Subject Classification
  The Mathematics Subject Classification is an alphanumerical classification scheme collaboratively produced by staff of and based on the coverage of the two major mathematical reviewing databases, Mathematical Reviews and Zentralblatt MATH...
  
   16Y60) (S,+,·) in which all elements have a multiplicative inverse. These objects are also called proper semifields. A variation of this definition arises if S contains an absorbing zero that is different from the multiplicative unit e, it is required that the non-zero elements be invertible, and a·0 = 0·a = 0. Since multiplication is associative, the (non-zero) elements of a semifield form a group
  Group (mathematics)
  In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
  
  . However, the pair (S,+) is only a semigroup
  Semigroup
  In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative binary operation. A semigroup generalizes a monoid in that there might not exist an identity element...
  
  , i.e. additive inverse need not exist, or, colloquially, 'there is no subtraction'. Sometimes, it is not assumed that the multiplication is associative.

Examples

We only give examples of semifields in the second sense, i.e. additive semigroups with distributive multiplication. Moreover, addition is commutative and multiplication is associative in our examples.

• Positive real number
  Real number
  In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
  
  s with the usual addition and multiplication form a commutative semifield.
• Rational function
  Rational function
  In mathematics, a rational function is any function which can be written as the ratio of two polynomial functions. Neither the coefficients of the polynomials nor the values taken by the function are necessarily rational.-Definitions:...
  
  s of the form f /g, where f and g are polynomial
  Polynomial
  In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
  
  s in one variable with positive coefficients form a commutative semifield.
• Max-plus algebra
  Max-plus algebra
  A max-plus algebra is an algebra over the real numbers with maximum and addition as the two binary operations.It can be used appropriately to determine marking times within a given Petri net and a vector filled with marking state at the beginning....
  
  , or the tropical semiring, (R, max, +) is a semifield. Here the sum of two elements is defined to be their maximum, and the product to be their ordinary sum.
• If (A,≤) is a lattice ordered group then (A,+,·) is an additively idempotent semifield. The semifield sum is defined to be the sup of two elements. Conversely, any additively idempotent semifield (A,+,·) defines a lattice-ordered group (A,≤), where a≤b if and only if a + b = b.