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.