In mathematics, a semifield is an

algebraic structure In mathematics, an algebraic structure 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 multiplication), and a finite set ...

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, addition and multiplication, which is similar to a

field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...

, 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 and finite geometry (

MSC MSC may refer to: Computers * Message Sequence Chart * Microelectronics Support Centre of UK Rutherford Appleton Laboratory * MIDI Show Control * MSC Malaysia (formerly known as Multimedia Super Corridor) * USB mass storage device class (USB MS ...

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 comm ...

, ** 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 operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures ...

, 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 or

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

. A semifield that is associative is a division ring, and one that is both associative and commutative is a

. 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 divisors, 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, functional analysis, and

theoretical computer science computer science (TCS) is a subset of general computer science and mathematics that focuses on mathematical aspects of computer science such as the theory of computation, lambda calculus, and type theory. It is difficult to circumscribe the ...

(

16Y60), a semifield is a semiring (''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 ide ...

. 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: ''x''·''y'', or simply ''xy'' ...

, 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 Positive is a property of positivity and may refer to: Mathematics and science * Positive formula, a logical formula not containing negation * Positive number, a number that is greater than 0 * Plus sign, the sign "+" used to indicate a posi ...

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

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 f ...

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. * Rational functions of the form ''f'' /''g'', where ''f'' and ''g'' are polynomials 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 measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...

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

semifield with the semifield sum defined to be the

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

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, and multiplication defined by

logical and In logic, mathematics and linguistics, And (\wedge) is the truth-functional operator of logical conjunction; the ''and'' of a set of operands is true if and only if ''all'' of its operands are true. The logical connective that represents this ...

References

{{Reflist, refs= Landquist, E.J., "On Nonassociative Division Rings and Projective Planes", Copyright 2000. Donald Knuth, ''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