Inmathematics, asemifield is analgebraic structure with twobinary operations, addition and multiplication, which is similar to afield, but with some axioms relaxed.

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

• Inprojective geometry andfinite geometry (MSC 51A, 51E, 12K10), asemifield is anonassociative division ring with multiplicative identity element.^[1] More precisely, it is anonassociative ring whose nonzero elements form aloop under multiplication. In other words, a semifield is a setS with two operations + (addition) and · (multiplication), such that
  • (S,+) is anabelian group,
  • multiplication isdistributive on both the left and right,
  • there exists a multiplicativeidentity element, and
  • division is always possible: for everya and every nonzerob inS, there exist uniquex andy inS for whichb·x =a andy·b =a.

Note in particular that the multiplication is not assumed to becommutative orassociative. A semifield that is associative is adivision ring, and one that is both associative and commutative is afield. A semifield by this definition is a special case of aquasifield. IfS is finite, the last axiom in the definition above can be replaced with the assumption that there are nozero divisors, so thata⋅b = 0 implies thata = 0 orb = 0.^[2] Note that due to the lack of associativity, the last axiom isnot equivalent to the assumption that every nonzero element has a multiplicative inverse, as is usually found in definitions of fields and division rings.

• Inring theory,combinatorics,functional analysis, andtheoretical computer science (MSC 16Y60), asemifield is asemiring (S,+,·) in which all nonzero elements have a multiplicative inverse.^[3][4] These objects are also calledproper semifields. A variation of this definition arises ifS contains an absorbing zero that is different from the multiplicative unite, it is required that the non-zero elements be invertible, anda·0 = 0·a = 0. Since multiplication isassociative, the (non-zero) elements of a semifield form agroup. However, the pair (S,+) is only asemigroup, i.e. additive inverse need not exist, or, colloquially, 'there is no subtraction'. Sometimes, it is not assumed that the multiplication is associative.

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.

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.

• Positiverational 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 theprobability semiring, which is isomorphic to thelog semiring.

• Rational functions of the formf /g, wheref andg arepolynomials over a subfield of real numbers in one variable with positive coefficients, form a commutative semifield.

  This can be extended to include 0.

• Thereal numbersR 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 thetropical semiring.

  This can be extended by −∞ (an absorbing 0); this is the limit (tropicalization) of thelog semiring as the base goes to infinity.

• Generalizing the previous example, if (A,·,≤) is alattice-ordered group then (A,+,·) is an additivelyidempotent semifield with the semifield sum defined to be thesupremum of two elements. Conversely, any additively idempotent semifield (A,+,·) defines a lattice-ordered group (A,·,≤), wherea≤b if and only ifa +b =b.
• The boolean semifieldB = {0, 1} with addition defined bylogical or, and multiplication defined bylogical and.

• Planar ternary ring (first sense)

• Algebraic structures
• Ring theory

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