# Abelian group

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**Abelian groups** generalize the arithmetic concept of addition of integers. (They were named after Niels Henrik Abel.^{[1]}).

- Thus, in abstract algebra, an
**abelian group**, is a**commutative group**, previously called a `symplectic' group.

An **abelian group** is defined as a set, **, together with an operation "" which is commutative, that is, for any elements and of **** one has that: **

The operation of addition of integers obviously has the property of commutativity.

## Notes[edit]

- 1. The collection of all Abelian groups, together with the group homomorphisms between them, forms the category
**, the prototype of an Abelian category.**

- 2. Almost all of the well-known algebraic structures other than Boolean algebras, are
*undecidable*(see Gödel's theorems; see also the following note). It was therefore rather surprising when one of Tarski's student, Szmielew proved in 1955 that the first order theory of abelian groups-- unlike its nonabelian counterpart-- is*decidable*. This important result provides an interesting link between the category of Abelian groups and that of Boolean algebras. - 3. Moreover, one expect that the theory of centered Łukasiewicz logic algebras should also be decidable because there is a fundamental, categorical adjointness defined between the category of
*centered*Łukasiewicz logic algebras and that of Boolean logic algebras^{[2]}. Note also that general Łukasiewicz n-valued logic algebras are `'noncommutative^{[3]}

## References[edit]

- ↑ Jacobson (2009), p. 41
- ↑ http://aux.planetphysics.us/files/papers/100/NvalLogicsGG.pdf Georgescu, G. 2006, N-valued Logics and Łukasiewicz-Moisil Algebras, Axiomathes, 16 (1-2): 123-136.
- ↑ http://images.planetphysics.us/files/lec/294/CategoryOfLMnLogicAlgebrasPP.pdf Topic: Algebraic category of Łukasiewicz-Moisil n-valued logic algebras.

- Jacobson, Nathan (2009).
*Basic Algebra I*(2nd ed.). Dover Publications. ISBN 978-0-486-47189-1. - Szmielew, Wanda (1955). "Elementary properties of abelian groups".
*Fundamenta Mathematicae***41**: 203–271. - Cox, David (2004).
*Galois Theory*. Wiley-Interscience. MR 2119052. - Fuchs, László (1970).
*Infinite Abelian Groups, Vol. I*. Pure and Applied Mathematics.**36-I**. Academic Press. MR 0255673.