On a special single-power cyclic hypergroup and its automorphisms

2016 ◽  
Vol 08 (04) ◽  
pp. 1650059 ◽  
Author(s):  
M. Al Tahan ◽  
B. Davvaz
Keyword(s):  
Abelian Group ◽  
Braid Group ◽  
Definition Of ◽  
Single Power

After introducing the definition of hypergroups by Marty, the study of hyperstructures and its applications has been of great importance. In this paper, we find a link between hyperstructures and the infinite non-abelian group, braid group [Formula: see text]. This is the first connection to be done between these two different domains. First, we define a new hyperoperation ⋆ associated to [Formula: see text] and study its properties. Next, we prove that [Formula: see text] is a single-power cyclic hypergroup with infinite period. Then, we define an onto homomorphism from [Formula: see text] to another hypergroup. Finally, we determine the set of all automorphisms of [Formula: see text] and prove that it is a group under the operation of functions composition.

scholarly journals A lattice-theoretic characterization of pure subgroups of Abelian groups

2021 ◽  
Author(s):  
M. Ferrara ◽  
M. Trombetti
Keyword(s):  
Abelian Group ◽  
Abelian Groups ◽  
Subgroup Lattice ◽  
General Definition ◽  
Short Paper ◽  
Theoretic Characterization ◽  
Definition Of

AbstractLet G be an abelian group. The aim of this short paper is to describe a way to identify pure subgroups H of G by looking only at how the subgroup lattice $$\mathcal {L}(H)$$ L ( H ) embeds in $$\mathcal {L}(G)$$ L ( G ) . It is worth noticing that all results are carried out in a local nilpotent context for a general definition of purity.

scholarly journals Planar pure braids on six strands

2020 ◽  
Vol 29 (01) ◽  
pp. 1950097
Author(s):  
Jacob Mostovoy ◽  
Christopher Roque-Márquez
Keyword(s):  
Abelian Group ◽  
Fundamental Group ◽  
Free Product ◽  
Free Group ◽  
Configuration Space ◽  
Braid Group ◽  
Free Abelian Group ◽  
Braid Groups ◽  
Pure Braid Group

The group of planar (or flat) pure braids on [Formula: see text] strands, also known as the pure twin group, is the fundamental group of the configuration space [Formula: see text] of [Formula: see text] labeled points in [Formula: see text] no three of which coincide. The planar pure braid groups on 3, 4 and 5 strands are free. In this note, we describe the planar pure braid group on 6 strands: it is a free product of the free group on 71 generators and 20 copies of the free abelian group of rank two.

scholarly journals A Generalization of Final Rank of Primary Abelian Groups

1970 ◽  
Vol 22 (6) ◽  
pp. 1118-1122 ◽  
Author(s):  
Doyle O. Cutler ◽  
Paul F. Dubois
Keyword(s):  
Abelian Group ◽  
Abelian Groups ◽  
Basic Subgroup ◽  
Primary Abelian Group ◽  
Limit Ordinal ◽  
Final Rank ◽  
Definition Of ◽  
Group Recall

Let G be a p-primary Abelian group. Recall that the final rank of G is infn∈ω{r(pnG)}, where r(pnG) is the rank of pnG and ω is the first limit ordinal. Alternately, if Γ is the set of all basic subgroups of G, we may define the final rank of G by supB∈Γ {r(G/B)}. In fact, it is known that there exists a basic subgroup B of G such that r(G/B) is equal to the final rank of G. Since the final rank of G is equal to the final rank of a high subgroup of G plus the rank of pωG, one could obtain the same information if the definition of final rank were restricted to the class of p-primary Abelian groups of length ω.

scholarly journals Local polynomial functions on factor rings of the integers

1979 ◽  
Vol 27 (2) ◽  
pp. 232-238 ◽  
Author(s):  
Hans Lausch ◽  
Wilfried Nöbauer
Keyword(s):  
Abelian Group ◽  
Universal Algebra ◽  
Polynomial Function ◽  
Commutative Rings ◽  
Infinite Length ◽  
Polynomial Functions ◽  
Local Polynomial ◽  
Definition Of ◽  
Image Position

AbstractLet A be a universal algebra. A function ϕ Ak-A is called a t-local polynomial function, if ϕ can ve interpolated on any t places of Ak by a polynomial function— for the definition of a polynomial function on A, see Lausch and Nöbauer (1973), Let Pk(A) be the set of the polynomial functions, LkPk(A) the set of all t-local polynmial functions on A and LPk(A) the intersection of all LtPk(A), then . If A is an abelian group, then this chain has at most five distinct members— see Hule and Nöbauer (1977)— and if A is a lattice, then it has at most three distinct members— see Dorninger and Nöbauer (1978). In this paper we show that in the case of commutative rings with identity there does not exist such a bound on the length of the chain and that, in this case, there exist chains of even infinite length.

scholarly journals BIRMAN'S CONJECTURE FOR SINGULAR BRAIDS ON CLOSED SURFACES

2004 ◽  
Vol 13 (07) ◽  
pp. 895-915
Author(s):  
LUIS PARIS
Keyword(s):  
Braid Group ◽  
Oriented Surface ◽  
Vassiliev Invariants ◽  
Closed Surfaces ◽  
Definition Of ◽  
Closed Oriented Surface

Let M be a closed oriented surface of genus g≥1, let Bn(M) be the braid group of M on n strings, and let SBn(M) be the corresponding singular braid monoid. Our purpose in this paper is to prove that the desingularization map η : SBn(M)→ℤ[Bn(M)], introduced in the definition of the Vassiliev invariants (for braids on surfaces), is injective.

scholarly journals On the dimension of an Abelian group

2021 ◽  
Vol 27 (4) ◽  
pp. 267-275
Author(s):  
Timo Tossavainen ◽  
◽  
Pentti Haukkanen ◽  
Keyword(s):  
Abelian Group ◽  
Dimensional Structure ◽  
Torsion Subgroup ◽  
Finitely Generated ◽  
Definition Of

We introduce a measure of dimensionality of an Abelian group. Our definition of dimension is based on studying perpendicularity relations in an Abelian group. For G ≅ ℤn, dimension and rank coincide but in general they are different. For example, we show that dimension is sensitive to the overall dimensional structure of a finite or finitely generated Abelian group, whereas rank ignores the torsion subgroup completely.

The braid group and its presentation

2021 ◽  
pp. 2150021
Author(s):  
Bronislaw Wajnryb
Keyword(s):  
Braid Group ◽  
Topological Proof ◽  
Definition Of ◽  
Standard Presentation ◽  
Geometric Definition

In this paper, we recall the geometric definition of the braid group by Emil Artin and we give a complete, elementary geometric/topological proof of the standard presentation of the braid group on [Formula: see text] strings.

Trace in additive categories

1970 ◽  
Vol 67 (3) ◽  
pp. 541-547 ◽  
Author(s):  
Alan Thomas
Keyword(s):  
Abelian Group ◽  
Commutative Diagram ◽  
Additive Category ◽  
Definition Of ◽  
Image Position

1. Let ℳ be an additive category. (We refer to ((1), Ch. IX) for the definition of an additive category and associated terms. In particular a sequenceis exact if i = kerp and p = coker i.) We write End M for Hom(M, M). A trace on ℳ with values in an abelian group G is a collection of (abelian group) homomorphismsone for each M ∈ ℳ, satisfying the following two conditions:(i) Exactness. Given a commutative diagram with exact rows,then tA(f) + tC(h) = tB(g).

The distribution of sequences in Abelian groups

1970 ◽  
Vol 67 (1) ◽  
pp. 1-11 ◽  
Author(s):  
J. S. Dennis
Keyword(s):  
Abelian Group ◽  
Uniform Distribution ◽  
Abelian Groups ◽  
Definition Of ◽  
Independent Distribution ◽  
Prime Exponent

1Introduction. In this paper, I give a definition of uniform distribution for sequences taking values in certain types of Abelian group—in particular, in groups of prime exponent—and one of independent distribution for sets of such sequences. In the various cases, I find conditions for the properties to hold, which are similar to Weyl's criterion for the uniform distribution of real sequences modulo 1.

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