scholarly journals A representation theorem for abelian groups with no elements of infinitep-height

1967 ◽  
Vol 20 (1) ◽  
pp. 31-33 ◽  
Author(s):  
Delmar Boyer ◽  
Adolf Mader
Keyword(s):  
Representation Theorem ◽  
Abelian Groups

A generalization of co-Hopfian abelian groups

2017 ◽  
Vol 27 (04) ◽  
pp. 351-360
Author(s):  
Andrey Chekhlov ◽  
Peter Danchev
Keyword(s):  
Abelian Group ◽  
Representation Theorem ◽  
Abelian Groups ◽  
Classical Notion ◽  
Group A ◽  
Hopfian Group ◽  
Comprehensive Study

We define the concept of an [Formula: see text]-co-Hopfian abelian group, which is a nontrivial generalization of the classical notion of a co-Hopfian group. A systematic and comprehensive study of these groups is given in very different ways. Specifically, a representation theorem for [Formula: see text]-co-Hopfian groups is established as well as it is shown that there exists an [Formula: see text]-co-Hopfian group which is not co-Hopfian.

scholarly journals The Hahn Embedding Theorem for a Class of Residuated Semigroups

Studia Logica ◽  
2020 ◽  
Vol 108 (6) ◽  
pp. 1161-1206
Author(s):  
Sándor Jenei
Keyword(s):  
Representation Theorem ◽  
Abelian Groups ◽  
Embedding Theorem ◽  
Residuated Lattices ◽  
Lexicographic Product ◽  
Product Construction

AbstractHahn’s embedding theorem asserts that linearly ordered abelian groups embed in some lexicographic product of real groups. Hahn’s theorem is generalized to a class of residuated semigroups in this paper, namely, to odd involutive commutative residuated chains which possess only finitely many idempotent elements. To this end, the partial lexicographic product construction is introduced to construct new odd involutive commutative residuated lattices from a pair of odd involutive commutative residuated lattices, and a representation theorem for odd involutive commutative residuated chains which possess only finitely many idempotent elements, by means of linearly ordered abelian groups and the partial lexicographic product construction is presented.

scholarly journals Inner Product Groups and Riesz Representation Theorem

Symmetry ◽  
2021 ◽  
Vol 13 (10) ◽  
pp. 1946
Author(s):  
Alireza Pourmoslemi ◽  
Tahereh Nazari ◽  
Mehdi Salimi
Keyword(s):  
Representation Theorem ◽  
Abelian Groups ◽  
Inner Product ◽  
Riesz Representation Theorem ◽  
Product Group ◽  
Basic Properties ◽  
Inner Products ◽  
Normed Group ◽  
Riesz Representation ◽  
Torsion Free

In this paper, we introduce an inner product on abelian groups and, after investigating the basic properties of the inner product, we first show that each inner product group is a torsion-free abelian normed group. We give examples of such groups and describe the norms induced by such inner products. Among other results, Hilbert groups, midconvex and orthogonal subgroups are presented, and a Riesz representation theorem on divisible Hilbert groups is proved.

Additive bases via Fourier analysis

2021 ◽  
pp. 1-12
Author(s):  
Bodan Arsovski
Keyword(s):  
Abelian Group ◽  
Fourier Analysis ◽  
Vector Space ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
Probabilistic Method ◽  
Additive Basis ◽  
Generating Sets

Abstract Extending a result by Alon, Linial, and Meshulam to abelian groups, we prove that if G is a finite abelian group of exponent m and S is a sequence of elements of G such that any subsequence of S consisting of at least $$|S| - m\ln |G|$$ elements generates G, then S is an additive basis of G . We also prove that the additive span of any l generating sets of G contains a coset of a subgroup of size at least $$|G{|^{1 - c{ \in ^l}}}$$ for certain c=c(m) and $$ \in = \in (m) < 1$$ ; we use the probabilistic method to give sharper values of c(m) and $$ \in (m)$$ in the case when G is a vector space; and we give new proofs of related known results.

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