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1

The zero-sum constant, the Davenport constant and their analogues

Zakarczemny Maciej

Technical Transactions

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2020

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Vol. 117, iss. 1

art. no. e2020027

EN

Let D(G) be the Davenport constant of a finite Abelian group G. For a positive integer m (the case m = 1, is the classical case) let Em(G) (or ηm(G)) be the least positive integer t such that every sequence of length t in G contains m disjoint zero‑sum sequences, each of length |G| (or of length ≤ exp(G), respectively). In this paper, we prove that if G is an Abelian group, then Em(G) = D(G) – 1 + m|G|, which generalizes Gao’s relation. Moreover, we examine the asymptotic behaviour of the sequences (Em(G))m≥1 and (ηm(G))m≥1. We prove a generalization of Kemnitz’s conjecture. The paper also contains a result of independent interest, which is a stronger version of a result by Ch. Delorme, O. Ordaz, D. Quiroz. At the end, we apply the Davenport constant to smooth numbers and make a natural conjecture in the non-Abelian case.

2

On m-ω1-pω+n - projective abelian p-groups

Danchev P.

Demonstratio Mathematica

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2014

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Vol. 47, nr 4

805--825

EN

For any non-negative integers m and n, we define the classes of m-ω1-pω+n -projective groups and strongly m-ω1-pω+n -projective groups, which properly encompass the classes of ω1-pω+n -projectives introduced by Keef in J. Algebra Numb. Th. Acad. (2010) and strongly ω1-pω+n -projectives introduced by the present author in Hacettepe J. Math. Stat. (2014), respectively. The new group structures share many interesting properties, which are closely related to these of the aforementioned two own subclasses. Moreover, certain basic results in this direction are also established.

3

Cyclic entropic quasigropups

Bińczak G., Kaleta J.

Demonstratio Mathematica

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2009

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Vol. 42, nr 2

269-281

EN

In this paper we explain the relationship of some entropic quasigroups to abelian groups with involution. It is known that (Zn, -n) are examples of cyclic entropic quasigroups which are not groups. We describe all cyclic entropic quasigroups with quasiidentity.

4

Full-rank non-periodic factorizations of elementary p-groups

Szabo S.

Demonstratio Mathematica

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2008

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Vol. 41, nr 1

45-56

EN

It will be shown that for each prime p there is a threshold value kp such that each elementary p-group of rank at least kp admits a non-periodic full-rank factorization. It will be established that k-2 < 15 and kp < 6 for p > 3.

5

Liczby niepewne

Borawski M.

Metody Informatyki Stosowanej

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2007

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nr 2(Tom 12)

27-34

EN

In the article, the origin of number systems, which describe the uncertain information is handled. Two different possibilities of creating such systems are shown. The convolution representation and its connection with fuzzy arithmetic and algebra is introduced. On this basis the need of defining a new conception of the certain number is proved. The new conception of such, numbers is defined in the article.

6

Comparability groups

Foulis D., Foulis D.

Demonstratio Mathematica

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2006

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Vol. 39, nr 1

15-32

EN

A comparability group is a unital group with a compression base and with the general comparability property. The additive group of self-adjoint elements in a von Neumann algebra, and any Dedekind sigma-complete lattice-ordered abelian group with order unit are examples of comparability groups. We develop the basic theory of comparability groups, and show that an archimedean comparability group with the Rickart projection property can be embedded in a partially ordered rational vector space the elements of which admit a rational spectral resolution.

7

An extension of a result of A. D. Sands

Szabo S.

Demonstratio Mathematica

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2000

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Vol. 33, nr 3

459-465

EN

A. D. Sands showed that if a group of type (22,22) is a direct product of its subsets of order 4, then at least one of these subsets must be periodic. In this paper we prove a result about groups of type (2\,2X) that generalizes Sands' theorem.

8

Homomorphisms of the group L1/2 into L1/2 and L1/3

Chudziak J., Drygaś P., Jabłoński W., Midura S.

Demonstratio Mathematica

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1998

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Vol. 31, nr 1

143-152

9

Elementary 3-groups with Hajos n-property

Szabo S.

Demonstratio Mathematica

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1998

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Vol. 31, nr 3

627-631

EN

Let G be an elementary 3-group . We characterize those n values for which from that G is a direct product of its subsets A1, . . . , An it follows that at least one Ai is a direct product of a nontrivial subgroup and a subset of G.