Semifield

Overview

• 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

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 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.