# Semifield

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In mathematics, a **semifield** is an algebraic structure with two binary operations, 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 and finite geometry (MSC 51A, 51E, 12K10), a
**semifield**is a nonassociative division ring with multiplicative identity element.^{[1]}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, - multiplication is distributive on both the left and right,
- there exists a multiplicative identity element, 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. A semifield that is associative is a division ring, 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 divisors, so that*a*·*b*= 0 implies that*a*= 0 or*b*= 0.^{[2]}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 (MSC 16Y60), a
**semifield**is a semiring (*S*,+,·) in which all nonzero elements have a multiplicative inverse.^{[3]}^{[4]}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. However, the pair (*S*,+) is only a semigroup, 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 with the usual addition and multiplication form a commutative semifield.
- This can be extended by an absorbing 0.

- Positive real numbers 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
**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 semifield with the semifield sum defined to be the supremum 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**= {0, 1} with addition defined by logical or, and multiplication defined by logical and.

## See also

- Planar ternary ring (first sense)

## References

- ↑ Donald Knuth,
*Finite semifields and projective planes*. J. Algebra, 2, 1965, 182--217 MR0175942. - ↑ Landquist, E.J., "On Nonassociative Division Rings and Projective Planes", Copyright 2000.
- ↑ 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, MR1163371. Kluwer Academic Publishers, Dordrecht, 1999. xii+381 pp. ISBN 0-7923-5786-8 MR1746739. - ↑ Hebisch, Udo; Weinert, Hanns Joachim,
*Semirings and semifields*. Handbook of algebra, Vol. 1, 425--462, North-Holland, Amsterdam, 1996. MR1421808.

Original source: https://en.wikipedia.org/wiki/Semifield.
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