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Concept
Semifield
Summary
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.
The term semifield has two conflicting meanings, both of which include fields as a special case.
In projective geometry and finite geometry (MSC 51A, 51E, 12K10), a semifield is a nonassociative division ring with multiplicative identity element. More precisely, it is a nonassociative ring 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 nonzero 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.
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