What is "semifield"

n. (context mathematics English) An algebraic structure with two binary operations, addition and multiplication, similar to the field but with some axioms relaxed.

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. 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 the analogue of a division algebra, but defined over the integers Z rather than over a field. More precisely, it is a Z-algebra 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. 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 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.