Half body

This article describes the term "half-body" as it is used in abstract algebra. For the coordinate range of the same name for planes in geometry, see Half-body (geometry) .

In algebra , especially in ring theory , a half body denotes the specialization of a half ring in which the multiplication not only forms a half group but a group . If the addition has a designated 0 element, the only requirement is that the multiplicative group extends over the elements other than 0.

Examples

The set of positive fractions together with the usual addition and multiplication form a half-body: ${\displaystyle \mathbb {Q} _{+}}$

• Addition and multiplication are both associative, so that the positive fractions under addition and multiplication each form at least a semi-group.
• Addition and multiplication are distributive, so that the positive fractions under addition and multiplication form a half ring.
• The positive fractions form a group under the multiplication, since the 1 (= 1/1) is positive and the reciprocal of each positive fraction is again a positive fraction.
• Without zero and without negative fractions, the neutral element and the inverse elements are missing with respect to the addition, so that the positive fractions do not form a group under the addition. ${\displaystyle \mathbb {Q} _{+}}$

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

Another example of a half-body is the whole numbers with the minimum operation (or maximum operation) as addition and the addition of whole numbers as multiplication. Because distributivity is via and fulfilled. ${\displaystyle \min(a,b)+c=\min(a+c,b+c)}$ ${\displaystyle c+\min(a,b)=\min(c+a,c+b)}$

Related structures

Analogous to the ring-like structures ring , fast ring , half-ring , there are the corresponding body-like structures oblique bodies , almost bodies and half bodies. In each of them, the multiplication only has to form a group (instead of just a semigroup) on elements other than 0. For the analog transition from ring to body, where multiplication is also required commutatively, there are no special analog terms, instead one simply says multiplicative commutative almost body / half body.

literature

• U. Hebisch; HJ Weinert: Half Rings - Algebraic Theory and Applications in Computer Science , Teubner, Stuttgart, 1993
• U. Hebisch; HJ Weinert: Semirings and Semifields. In Handbook of Algebra. Elsevier, 1996.