Torsion-free abelian group

About: Torsion-free abelian group is a research topic. Over the lifetime, 71 publications have been published within this topic receiving 3099 citations.

Invariant decomposition of functions with respect to commuting invertible transformations

Abstract: Consider a 1 ,..., , an ∈ R arbitrary elements. We characterize those functions f: R→ R that decompose into the sum of a j -periodic functions, i.e., f=f 1 +---+f n with Δ aJ f(x):= f(x+aj)-f(x) = 0. We showthat f has such a decomposition if and only if for all partitions Β 1 UB 2 U...US N = {a 1 ,..., a n ) } with Bj consisting of commensurable elements with least common multiples b j one has Δ b1 ... Δb N f = 0. Actually, we prove a more general result for periodic decompositions of functions f: A→ R defined on an Abelian group A; in fact, we even consider invariant decompositions of functions f: A → R with respect to commuting, invertible self-mappings of some abstract set A. We also extend our results to functions between torsion free Abelian groups. As a corollary we also obtain that on a torsion free Abelian group the existence of a real-valued periodic decomposition of an integer-valued function implies the existence of an integer-valued periodic decomposition with the same periods.

Invariant decomposition of functions with respect to commuting invertible transformations

Abstract: Consider a_1,a_2,...,a_n, arbitrary elements of R. We characterize those real functions f that decompose into the sum of a_j-periodic functions, i.e., f=f_1+...+f_n with D_{a_j}f(x):=f(x+a_j)-f(x)=0. We show that f has such a decomposition if and only if for all partitions to B_1, B_2,... B_N of {a_1,a_2,...,a_n} with B_j consisting of commensurable elements with least common multiples b_j, one has D_{b_1}... D_{b_N}f=0. Actually, we prove a more general result for periodic decompositions of real functions f defined on an Abelian group A, and, in fact, we even consider invariant decompositions of functions f defined on some abstract set A, with respect to commuting, invertible self-mappings of the set A. We also extend our results to functions between torsion free Abelian groups. As a corollary we also obtain that on a torsion free Abelian group the existence of a real valued periodic decomposition of an integer valued function implies the existence of an integer valued periodic decomposition with the same periods.

Wiener-Hopf operators on a subsemigroup of a discrete torsion free abelian group

Abstract: In the paper Wiener-Hopf operators on a semigroup of nonnegative elements of a linearly quasi-ordered torsion free Abelian group are considered. Wiener-Hopf factorization of an invertible element of the group algebra is constructed, notions of a topological index and a factor index are introduced. It turns out that the set of factor indices for invertible elements of the group algebra is a linearly ordered group. It is shown that Wiener-Hopf operator with an invertible symbol is an one-side invertible operator and its invertibility properties are defined by the sign of the factor index of its symbol. Groups on which there exist nontrivial Fredholm Wiener-Hopf operators are described. As an example, all linear quasi-orders on the group ℤn are found and corresponding Wiener-Hopf operators are considered.

Formal power series representations of free exponential groups

Abstract: A classical result of Magnus asserts that a free group F has a faithful representation in the group of units of a ring of non-commuting formal power series with integral coefficients. We generalize this result to the category of A-groups, where A is an associative ring or an Abelian group. We say that a free A-group FA has a faithful Magnus representation if there is a ring B containing A as an additive subgroup (or a subring) such that FA is faithfully represented (exactly as in Magnus' classical result in the case A = Z)in the group of units of the ring of formal power series in non-communting indeterminater over B.The three principal results are: (I) If A is a torsion free Abelian group and FA is a free A-groupp of Lyndon' type, then FA has a faithful Magnus representation; (II) If A is an ω‐residually Z ring, then FA has a faithful Magnus representation;(III) for every nontrivial torsion-free Abelian group A, FA has a faithful Magnus representation in B[[Y]] for a suitable ring B in and only if FQ has...