Semifield

In mathematics, a semifield is an algebraic structure with two binary operations, addition and multiplication, which is similar to the field, but with some axioms relaxed. There are at least two conflicting conventions of what constitutes a semifield.

• In projective geometry and finite geometry (MSC 51A, 51E, 12K10), a semifield is a ring (S,+,·) where (S,+) is an abelian group with identity element 0, the multiplication is distributive with respect to the addition on the left and on the right, and (S,·) is a division ring that is not assumed to be commutative or associative. This structure is a special case of a quasifield. 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, combinatorics, functional analysis, and theoretical computer science, a semifield is a semiring (MSC 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. 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.

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 numbers with the usual addition and multiplication form a commutative semifield.
• Rational functions of the form f /g, where f and g are polynomials in one variable with positive coefficients form a commutative semifield.
• Max-plus algebra, 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.

References

• Donald Knuth, Finite semifields and projective planes. J. Algebra, 2, 1965, 182--217

• 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