Abelian group

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, $A$ , together with an operation " $*$ " which is commutative, that is, for any elements $x$ and $y$ of $A$ one has that:
$x*y = y*x.$

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

Notes

1. The collection of all Abelian groups, together with the group homomorphisms between them, forms the category $Ab$ , 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

↑ 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.
Fuchs, László (1970). Infinite Abelian Groups, Vol. I. Pure and Applied Mathematics. 36-I. Academic Press.

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