semifield

A semifield is an algebraic structure that generalizes a field by relaxing some of the usual associative or commutative requirements. Formally, it is a set S equipped with two binary operations, addition and multiplication, such that:

- (S, +) is an abelian group with additive identity 0.

- There exists a multiplicative identity 1 ≠ 0, so 1·a = a·1 = a for all a in S.

- Multiplication distributes over addition from both sides: a·(b+c) = a·b + a·c and (a+b)·c = a·c + b·c for all a, b, c in S.

- For every a ≠ 0, the maps x ↦ a·x and x ↦ x·a are bijections (i.e., left and right division by a is always possible). This allows solving equations a·x = y and x·a = y uniquely for any y.

Multiplication in a semifield need not be associative or commutative. If multiplication happens to be associative and commutative, the semifield is a field. Thus semifields generalize fields by dropping some algebraic laws.

Finite semifields are particularly important in finite geometry, where they provide coordinates for translation planes. In this context, a semifield defines a non-Desarguesian plane when nonassociative, while the Desarguesian (field) case corresponds to the standard projective planes.

Examples include the ordinary finite fields, which are associative and commutative semifields, and many nonassociative constructions studied in finite geometry, such as Knuth semifields and various families built from finite fields with twisted or modified multiplications. Some authors distinguish semantically between semifields and related notions like presemifields, the latter lacking a multiplicative identity.