mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a semifield is an

algebraic structure In mathematics, an algebraic structure or algebraic system consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplicatio ...

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, a binary operation ...

s, addition and multiplication, which is similar to a field, but with some axioms relaxed.

Overview

The term semifield has two conflicting meanings, both of which include fields as a special case. * In

projective geometry In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting (''p ...

and

finite geometry A finite geometry is any geometry, geometric system that has only a finite set, finite number of point (geometry), points. The familiar Euclidean geometry is not finite, because a Euclidean line contains infinitely many points. A geometry based ...

( MSC 51A, 51E, 12K10), a semifield is a nonassociative division ring with multiplicative identity element. More precisely, it is a nonassociative ring whose nonzero elements form a loop under multiplication. In other words, a semifield is a set ''S'' with two operations + (addition) and · (multiplication), such that ** (''S'',+) is an

abelian group In mathematics, 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 the order in which they are written. That is, the group operation is commu ...

, ** multiplication is distributive on both the left and right, ** there exists a multiplicative

identity element In mathematics, an identity element or neutral element of a binary operation is an element that leaves unchanged every element when the operation is applied. For example, 0 is an identity element of the addition of real numbers. This concept is use ...

, and ** division is always possible: for every ''a'' and every nonzero ''b'' in ''S'', there exist unique ''x'' and ''y'' in ''S'' for which ''b''·''x'' = ''a'' and ''y''·''b'' = ''a''. : Note in particular that the multiplication is not assumed to be

commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Perhaps most familiar as a pr ...

associative In mathematics, the associative property is a property of some binary operations that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement for express ...

. A semifield that is associative is a

division ring In algebra, a division ring, also called a skew field (or, occasionally, a sfield), is a nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial ring in which every nonzero element has a multiplicativ ...

, and one that is both associative and commutative is a field. A semifield by this definition is a special case of a quasifield. If ''S'' is finite, the last axiom in the definition above can be replaced with the assumption that there are no

zero divisor In abstract algebra, an element of a ring is called a left zero divisor if there exists a nonzero in such that , or equivalently if the map from to that sends to is not injective. Similarly, an element of a ring is called a right ze ...

s, so that ''a''⋅''b'' = 0 implies that ''a'' = 0 or ''b'' = 0. Note that due to the lack of associativity, the last axiom is ''not'' equivalent to the assumption that every nonzero element has a multiplicative inverse, as is usually found in definitions of fields and division rings. * In ring theory,

combinatorics Combinatorics is an area of mathematics primarily concerned with counting, both as a means and as an end to obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many ...

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 (for example, Inner product space#Definition, inner product, Norm (mathematics ...

, and

theoretical computer science Theoretical computer science is a subfield of computer science and mathematics that focuses on the Abstraction, abstract and mathematical foundations of computation. It is difficult to circumscribe the theoretical areas precisely. The Associati ...

( MSC 16Y60), a semifield is a

semiring In abstract algebra, a semiring is an algebraic structure. Semirings are a generalization of rings, dropping the requirement that each element must have an additive inverse. At the same time, semirings are a generalization of bounded distribu ...

(''S'',+,·) in which all nonzero 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

, the (non-zero) elements of a semifield form a

group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic iden ...

. However, the pair (''S'',+) is only a

semigroup In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplicatively (just notation, not necessarily th ...

, i.e. additive inverse need not exist, or, colloquially, 'there is no subtraction'. Sometimes, it is not assumed that the multiplication is associative.

Primitivity of semifields

A semifield D is called right (resp. left) primitive if it has an element w such that the set of nonzero elements of D* is equal to the set of all right (resp. left) principal powers of w.

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

rational numbers In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (for examp ...

with the usual addition and multiplication form a commutative semifield. *:This can be extended by an absorbing 0. *

Positive real numbers In mathematics, the set of positive real numbers, \R_ = \left\, is the subset of those real numbers that are greater than zero. The non-negative real numbers, \R_ = \left\, also include zero. Although the symbols \R_ and \R^ are ambiguously used fo ...

with the usual addition and multiplication form a commutative semifield. *:This can be extended by an absorbing 0, forming the probability semiring, which is isomorphic to the

log semiring In mathematics, in the field of tropical analysis, the log semiring is the semiring structure on the logarithmic scale, obtained by considering the extended real numbers as logarithms. That is, the operations of addition and multiplication are def ...

. *

Rational function In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be ...

s of the form ''f'' /''g'', where ''f'' and ''g'' are

polynomial In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...

s over a subfield of real numbers in one variable with positive coefficients, form a commutative semifield. *:This can be extended to include 0. * The

real numbers In mathematics, a real number is a number that can be used to measurement, measure a continuous variable, continuous one-dimensional quantity such as a time, duration or temperature. Here, ''continuous'' means that pairs of values can have arbi ...

R can be viewed a semifield where the sum of two elements is defined to be their maximum and the product to be their ordinary sum; this semifield is more compactly denoted (R, max, +). Similarly (R, min, +) is a semifield. These are called the

tropical semiring In idempotent analysis, the tropical semiring is a semiring of extended real numbers with the operations of minimum (or maximum) and addition replacing the usual ("classical") operations of addition and multiplication, respectively. The tropical s ...

. *:This can be extended by −∞ (an absorbing 0); this is the limit ( tropicalization) of the

as the base goes to infinity. * Generalizing the previous example, if (''A'',·,≤) is a lattice-ordered group then (''A'',+,·) is an additively

idempotent Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of pl ...

semifield with the semifield sum defined to be the

supremum In mathematics, the infimum (abbreviated inf; : infima) of a subset S of a partially ordered set P is the greatest element in P that is less than or equal to each element of S, if such an element exists. If the infimum of S exists, it is unique, ...

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''. * The boolean semifield B = with addition defined by

logical or In logic, disjunction (also known as logical disjunction, logical or, logical addition, or inclusive disjunction) is a logical connective typically notated as \lor and read aloud as "or". For instance, the English language, English language ...

, and multiplication defined by

logical and In logic, mathematics and linguistics, ''and'' (\wedge) is the truth-functional operator of conjunction or logical conjunction. The logical connective of this operator is typically represented as \wedge or \& or K (prefix) or \times or \cdo ...

References

Donald Knuth Donald Ervin Knuth ( ; born January 10, 1938) is an American computer scientist and mathematician. He is a professor emeritus at Stanford University. He is the 1974 recipient of the ACM Turing Award, informally considered the Nobel Prize of comp ...

, ''Finite semifields and projective planes''. J. Algebra, 2, 1965, 182--217 {{MathSciNet, id=0175942. 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, {{MathSciNet, id=1163371. Kluwer Academic Publishers, Dordrecht, 1999. xii+381 pp. {{isbn, 0-7923-5786-8 {{MathSciNet, id=1746739. Hebisch, Udo; Weinert, Hanns Joachim, ''Semirings and semifields''. Handbook of algebra, Vol. 1, 425--462, North-Holland, Amsterdam, 1996. {{MathSciNet, id=1421808. Algebraic structures Ring theory

Overview

Primitivity of semifields

Examples

See also

References