7/18/13

Groups & Bodies.

Group is nonempty set with inner 2-argument operation 'o' with following conditions (axioms):

1.

For each a,b,c in G: (a o b) o c = a o (b o c)

2.

Exists e in G that for each a in G: a o e = a = e o a

3.

For each a in G exists a' in G that: a o a' = e = a' o a

if also:

For each a,b in G: a o b = b o a then we call it, abelian group.

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Body (or more strictly Commutative Body) is set K with at least 2 elements, with 2 2-argument inner operations, addition (+), and multiplication ('*'), when following conditions (axioms):

1.

{K, +} is abelian group (with neutral element is denoted as 1, and opposite element to a as -a.

2.

{ K - {0}, * } is abelian group (with neutral element is denoted as 1, and opposite element to a as a^-1),

3.

For each a,b,c in K: a * (b+c) = a*b + a*c.


Source: [1].

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