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.

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.

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.

See also

References

  1. ^ Donald Knuth, Finite semifields and projective planes. J. Algebra, 2, 1965, 182--217 MR0175942.
  2. ^ Landquist, E.J., "On Nonassociative Division Rings and Projective Planes", Copyright 2000.
  3. ^ 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.
  4. ^ Hebisch, Udo; Weinert, Hanns Joachim, Semirings and semifields. Handbook of algebra, Vol. 1, 425--462, North-Holland, Amsterdam, 1996. MR1421808.