scholarly journals Algebraic structure of countably compact non-torsion Abelian groups of size continuum from selective ultrafilters

2021 ◽  
Vol 297 ◽  
pp. 107703
Author(s):  
Matheus Koveroff Bellini ◽  
Ana Carolina Boero ◽  
Vinicius de Oliveira Rodrigues ◽  
Artur Hideyuki Tomita
2019 ◽  
Vol 267 ◽  
pp. 106894
Author(s):  
Matheus Koveroff Bellini ◽  
Ana Carolina Boero ◽  
Irene Castro-Pereira ◽  
Vinicius de Oliveira Rodrigues ◽  
Artur Hideyuki Tomita

Author(s):  
Amaira Moaitiq Mohammed Al-Johani

In abstract algebra, an algebraic structure is a set with one or more finitary operations defined on it that satisfies a list of axioms. Algebraic structures include groups, rings, fields, and lattices, etc. A group is an algebraic structure (????, ∗), which satisfies associative, identity and inverse laws. An Abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, these are the groups that obey the axiom of commutatively. The concept of an Abelian group is one of the first concepts encountered in abstract algebra, from which many other basic concepts, such as rings, commutative rings, modules and vector spaces are developed. This study sheds the light on the structure of the finite abelian groups, basis theorem, Sylow’s theorem and factoring finite abelian groups. In addition, it discusses some properties related to these groups. The researcher followed the exploratory and comparative approaches to achieve the study objective. The study has shown that the theory of Abelian groups is generally simpler than that of their non-abelian counter parts, and finite Abelian groups are very well understood.  


10.37236/2676 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
William E. Clark ◽  
Xiang-dong Hou

For each pointed abelian group $(A,c)$, there is an associated Galkin quandle $G(A,c)$ which is an algebraic structure defined on $\Bbb Z_3\times A$ that can be used to construct knot invariants. It is known that two finite Galkin quandles are isomorphic if and only if their associated pointed abelian groups are isomorphic. In this paper we classify all finite pointed abelian groups. We show that the number of nonisomorphic pointed abelian groups of order $q^n$ ($q$ prime) is $\sum_{0\le m\le n}p(m)p(n-m)$, where $p(m)$ is the number of partitions of integer $m$.


Author(s):  
MICHEL GRABISCH ◽  
BERNARD DE BAETS ◽  
JÁNOS FODOR

We consider the interval ]-1, 1[ and intend to endow it with an algebraic structure like a ring. The motivation lies in decision making, where scales that are symmetric w.r.t. 0 are needed in order to represent a kind of symmetry in the behaviour of the decision maker. A former proposal due to Grabisch was based on maximum and minimum. In this paper, we propose to build our structure on t-conorms and t-norms, and we relate this construction to uninorms. We show that the only way to build a group is to use strict t-norms, and that there is no way to build a ring. Lastly, we show that the main result of this paper is connected to the theory of ordered Abelian groups.


1981 ◽  
Vol 33 (3) ◽  
pp. 664-670 ◽  
Author(s):  
M. A. Khan

In [4], Edwin Hewitt denned a-rich LCA (i.e., locally compact abelian) groups and classified them by their algebraic structure. In this paper, we study LCA groups with some properties related to a-richness. We define an LCA group G to be power-rich if for every open neighbourhood V of the identity in G and for every integer n > 1, λ(nV) > 0, where nV = {nx ∈ G : x ∈ V} and λ is a Haar measure on G. G is power-meagre if for every integer n > 1, there is an open neighbourhood V of the identity, possibly depending on n, such that λ(nV) = 0. G is power-deficient if for every integer n > 1 and for every open neighbourhood V of the identity such that is compact, . G is dual power-rich if both G and Ĝ are power-rich. We define dual power-meagre and dual power-deficient groups similarly.


2018 ◽  
Vol 27 (14) ◽  
pp. 1850064 ◽  
Author(s):  
Khaled Bataineh ◽  
Mohamed Elhamdadi ◽  
Mustafa Hajij ◽  
William Youmans

We give a generating set of the generalized Reidemeister moves for oriented singular links. We then introduce an algebraic structure arising from the axiomatization of Reidemeister moves on oriented singular knots. We give some examples, including some non-isomorphic families of such structures over non-abelian groups. We show that the set of colorings of a singular knot by this new structure is an invariant of oriented singular knots and use it to distinguish some singular links.


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