Semifield

In algebra, particularly ring theory, a half-body denotes the specialization of a half-ring in which the multiplication is not only a semigroup, but a group. Does the addition of a designated 0 - element is only required that the multiplicative group extends 0 different elements of the of the.

Examples

The set of positive fractions along with the usual addition and multiplication forms a half-body:

• Addition and multiplication are both associative, so that the positive fractions under addition and multiplication to form at least half each group.
• Addition and multiplication are distributive, so that the positive fractions form a semiring under addition and multiplication.
• Positive fractions to form a group of the multiplication as the 1 (= 1/1 ) is positive and the inverse of each positive breakage is a positive fraction again.
• Excluding zero, and no negative fractions lacking the neutral element, and the inverse element with respect to addition, so that the openings do not form a positive group of the addition.

By adding the zero and negative rational numbers, the positive fractions can be extended to a body.

Another example of a half-body are integers with the minimum operation (or maximum ) operation as an addition, and the addition of integers as a multiplication. Distributivity is because the via min ( a, b ​​) c = min ( a c, b c) and c min ( A, B) = min ( c a, c b ) is satisfied.

Related structures

Similar to the ring-like structures ring, fast ring, half-ring, there is a corresponding body-like structures skew field, Fast body and half body. In them, the multiplication has only one (instead of just a semigroup ) form on the various elements of 0, a group. For the analog transition ring on the body, where the multiplication is also called commutative, there is no special analog terms, instead simply says multiplicatively commutative Nearly body / half body.