If we consider maps satisfying the same properties, but we allow to replace the real numbers with an arbitrary

abelian group, we come to the more general notion of valuation.

So, an element u [member of] [R.sup.x] is a unit of R if there exist u' [member of] [R.sup.x] such that uu' = u'u = 1, and [R.sup.x] forms an

abelian group under usual multiplication of R.

This math text explains the origins and underlying assumptions of the Cantor set, an ordered

abelian group. Chapters are devoted to the Bratteli-Vershik model, the Effros-Handelman-Shen theorem, the Bratelli-Elliott-Krieger theorem, measure theory, and the absorption theorem.

In the case that X is a commutative ring or a finite

abelian group (written multiplicative) and f maps x into its k-th power in X, we denote G(X, f) by G(X, k).

Every finite

Abelian group is determined by its endomorphism monoid in the class of all groups.

Let G be an

abelian group and [alpha] be a fixed nonzero element of G.

Then G/G' is an

abelian group and G' is minimal with respect to this property.

Let H be a finite

abelian group written additively and End(H) be the endomorphism ring of H.

More precisely, for a field k, a joint determinant D (= [D.sub.l]) (l [greater than or equal to] 1) is defined as a map from the set of l-tuples of commuting matrices in [GL.sub.n](k) (n [greater than or equal to] 1) into some

abelian group (G, +) which satisfies the following properties.

The infinite-dimensional torus [OMEGA] with the product topology and operation of pointwise multiplication is a compact topological

Abelian group. Therefore, on ([OMEGA], b([OMEGA])), where b(X) is the Borel [sigma]-field of the space X, the probability Haar measure [m.sub.H] exists, and this gives the probability space ([OMEGA], b([OMEGA]), [m.sub.H]).

Now consider the

abelian group hom(G, [S.sup.1]) of continuous homomorphisms from G to [S.sup.1] endowed with the group structure given by pointwise multiplication.

In order to study self-dual abelian codes, it is therefore restricted to the group algebra [mathematical expression not reproducible], where A is an

abelian group of odd order and B is a nontrivial

abelian group of two power order.