Semifield
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.
Overview
The term semifield has two conflicting meanings, both of which include fields as a special case.- In projective geometry and finite geometry, 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 + and ·, such that
- * 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''.
- In ring theory, combinatorics, functional analysis, and theoretical computer science, a semifield is a semiring 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 elements of a semifield form a group. However, the pair 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.
Primitivity of semifields
A semifield D is called right primitive if it has an element w such that the set of nonzero elements of D* is equal to the set of all right principal powers of w.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 rational 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 the probability semiring, which is isomorphic to the log semiring.
- Rational functions of the form f /g, where f and g are polynomials over a subfield of real numbers in one variable with positive coefficients, form a commutative semifield.
- :This can be extended to include 0.
- The real numbers R 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. Similarly is a semifield. These are called the tropical semiring.
- :This can be extended by −∞ ; this is the limit of the log semiring as the base goes to infinity.
- Generalizing the previous example, if is a lattice-ordered group then is an additively idempotent semifield with the semifield sum defined to be the supremum of two elements. Conversely, any additively idempotent semifield defines a lattice-ordered group, where a≤''b if and only if a'' + b = b.
- The Boolean semifield B = with addition defined by logical or, and multiplication defined by logical and.