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Abelian groups with the direct summand sum property

Vălcan D.

Demonstratio Mathematica

2002

Vol. 35, nr 3

477-491

If R is an associative ring, with unity, the R- module (the abelian group) M is said to have the direct summand sum property (in short D.S.S.P.) if the sum (that is the submodule (the subgroup) of M generated by the union) of any two direct summands ofM is again a direct summand in M. The present work gives descriptions of some classes of abelian groups with this property.

On subclasses of groups without free subsemigroups

Bajorska B., Macedońska O.

Demonstratio Mathematica

2000

Vol. 33, nr 1

35-42

The paper is inspired by the question of A. Shalev about possible coincidence of the class of collapsing groups and groups satisfying positive laws. We split the class of collapsing groups for subclasses, corresponding to different functions on natural numbers and give a positive answer for some of them.

On Certain Subclasses of the Classes Lc

Rajba T.

Probability and Mathematical Statistics

1999

Vol. 19, Fasc. 1

171--180

Loève in [5] introduced the classes LC associated with number c, c ϵ R, as the classes of probability measures satisfying the condition (1). Many authors investigated those classes ([2], [5]-[9], [20], [21]). In this paper we consider certain subclasses Lc1,…,ck, Lc1(k) of the classes Lc. We prove that they coincide with the classes of distributions of series of some random variables and with the classes of limit distributions of some normed sums. We give a characterization of certain classes Dc1,…,ck associated with Lc1,…,ck. Urbanik in [18] introduced the concept of the decomposability semigroup associated with probability measure P, as the set of all numbers c, such that P ϵ LC ([11]-[14]). The class L of self-decomposable distributions coincides with the class of probability measures P such that D(P) ⸧ [0, 1]. The class Lm, m ≥ 1, of multiply selfdecomposable distributions may be described as the class of probability measures P such that P ϵ Lc1,…,cm, for every c1,…, cm ϵ [0, 1], or in terms of multiply decomposability semigroups it is equivalent to the inclusion Dm (P) ⸧ [0, 1]m, where Dm (P) is the multiply decomposability semigroup defined by the formula Dm (P) = {(c1,…, cm); P ϵ Lc1,…,cm} ([3], [4], [10], [15]-[17], [19]).