scholarly journals Definition and Properties of Direct Sum Decomposition of Groups

2015 ◽  
Vol 23 (1) ◽  
pp. 15-27
Author(s):  
Kazuhisa Nakasho ◽  
Hiroshi Yamazaki ◽  
Hiroyuki Okazaki ◽  
Yasunari Shidama
Keyword(s):  
Direct Product ◽  
Equivalent Form ◽  
Fourth Section ◽  
Product Group ◽  
The Third ◽  
Direct Sum ◽  
Direct Sum Decomposition ◽  
Direct Product Group ◽  
Product Map

Summary In this article, direct sum decomposition of group is mainly discussed. In the second section, support of element of direct product group is defined and its properties are formalized. It is formalized here that an element of direct product group belongs to its direct sum if and only if support of the element is finite. In the third section, product map and sum map are prepared. In the fourth section, internal and external direct sum are defined. In the last section, an equivalent form of internal direct sum is proved. We referred to [23], [22], [8] and [18] in the formalization.

scholarly journals Equivalent Expressions of Direct Sum Decomposition of Groups

2015 ◽  
Vol 23 (1) ◽  
pp. 67-73
Author(s):  
Kazuhisa Nakasho ◽  
Hiroyuki Okazaki ◽  
Hiroshi Yamazaki ◽  
Yasunari Shidama
Keyword(s):  
Equivalent Form ◽  
Normal Subgroups ◽  
Direct Sum ◽  
Direct Sum Decomposition ◽  
Equivalent Expressions

Summary In this article, the equivalent expressions of the direct sum decomposition of groups are mainly discussed. In the first section, we formalize the fact that the internal direct sum decomposition can be defined as normal subgroups and some of their properties. In the second section, we formalize an equivalent form of internal direct sum of commutative groups. In the last section, we formalize that the external direct sum leads an internal direct sum. We referred to [19], [18] [8] and [14] in the formalization.

scholarly journals Locally adequate semigroup algebras

2016 ◽  
Vol 14 (1) ◽  
pp. 29-48 ◽  
Author(s):  
Yingdan Ji ◽  
Yanfeng Luo
Keyword(s):  
Direct Product ◽  
Simple Semigroup ◽  
Semigroup Algebra ◽  
Semigroup Algebras ◽  
Indecomposable Modules ◽  
Direct Sum ◽  
Multiplicative Basis ◽  
Special Cases ◽  
Direct Sum Decomposition ◽  
Adequate Semigroup

AbstractWe build up a multiplicative basis for a locally adequate concordant semigroup algebra by constructing Rukolaĭne idempotents. This allows us to decompose the locally adequate concordant semigroup algebra into a direct product of primitive abundant $0{\rm{ - }}{\cal J}*$-simple semigroup algebras. We also deduce a direct sum decomposition of this semigroup algebra in terms of the ${\cal R}*$-classes of the semigroup obtained from the above multiplicative basis. Finally, for some special cases, we provide a description of the projective indecomposable modules and determine the representation type.

The Reduction of Differential Calculus on Direct Product Group

1998 ◽  
Vol 30 (2) ◽  
pp. 257-262
Author(s):  
Feng Bo ◽  
Li Jianming ◽  
Song Xingchang
Keyword(s):  
Direct Product ◽  
Differential Calculus ◽  
Product Group ◽  
Direct Product Group

scholarly journals On Pierce-like idempotents and Hopf invariants

2003 ◽  
Vol 2003 (62) ◽  
pp. 3903-3920
Author(s):  
Giora Dula ◽  
Peter Hilton
Keyword(s):  
Homotopy Group ◽  
Hopf Invariant ◽  
The Third ◽  
Direct Sum ◽  
Direct Sum Decomposition ◽  
Matrix Calculus ◽  
Hopf Invariants

Given a setKwith cardinality‖K‖ =n, a wedge decomposition of a spaceYindexed byK, and a cogroupA, the homotopy groupG=[A,Y]is shown, by using Pierce-like idempotents, to have a direct sum decomposition indexed byP(K)−{ϕ}which is strictly functorial ifGis abelian. Given a classρ:X→Y, there is a Hopf invariantHIρon[A,Y]which extends Hopf's definition whenρis a comultiplication. ThenHI=HIρis a functorial sum ofHILoverL⊂K,‖L‖ ≥2. EachHILis a functorial composition of four functors, the first depending only onAn+1, the second only ond, the third only onρ, and the fourth only onYn. There is a connection here with Selick and Walker's work, and with the Hilton matrix calculus, as described by Bokor (1991).

IMPLEMENTATION OF ELECTRONIC MONITORING OF SENTENCED PERSONS IN THE SLOVAK REPUBLIC

2020 ◽  
Vol 10 (5) ◽  
pp. 59-75
Author(s):  
JAROSLAV KLÁTIK ◽  
◽  
LIBOR KLIMEK
Keyword(s):  
Restorative Justice ◽  
Electronic Monitoring ◽  
Slovak Republic ◽  
Pilot Project ◽  
Legal Regulation ◽  
European Level ◽  
Criminal Proceedings ◽  
Council Of Europe ◽  
Fourth Section ◽  
The Third

The work deals with implementation of electronic monitoring of sentenced persons in the Slovak Republic. It is divided into eight sections. The first section introduces restorative justice as a prerequisite of electronic monitoring in criminal proceedings. While the second section points out at the absence of legal regulation of electronic monitoring of sentenced persons at European level, the third section points out at recommendations of the Council of Europe addressed to European States. The fourth section analyses relevant alternative punishments in Slovak criminal justice. The fifth section introduces early beginnings of implementation of concerned system - the pilot project “Electronic Personnel Monitoring System” of the Ministry of Justice of the Slovak Republic. While the sixth section is focused on Slovak national law regulating electronic monitoring of sentenced persons - the Act No. 78/2015 Coll. on Control of the Enforcement of Certain Decisions by Technical Instruments, the seventh section is focused on further amendments of Slovak national law - namely the Act No. 321/2018 Coll. and the Act No. 214/2019 Coll. The last eight section introduces costs of system implementation and its operation.

Logicism Reconsidered

2009 ◽  
Author(s):  
Agustín Rayo
Keyword(s):  
Fourth Section ◽  
Truth Values ◽  
The Third

This article is divided into four sections. The first two identify different logicist theses, and show that their truth-values can be established given minimal assumptions. The third section sets forth a notion of “content-recarving” as a possible constraint on logicist theses. The fourth section—which is largely independent from the rest of the article—is a discussion of “neologicism.”

Epistemic self-esteem of philosophers in the face of philosophical disagreement

Human Affairs ◽  
2020 ◽  
Vol 30 (3) ◽  
pp. 328-342
Author(s):  
László Bernáth ◽  
János Tőzsér
Keyword(s):  
Self Esteem ◽  
Moderate Level ◽  
Significant Extent ◽  
Fourth Section ◽  
Philosophical Beliefs ◽  
Low Level ◽  
The Third ◽  
Epistemic Peers ◽  
The Face ◽  
And Performance

AbstractOur paper consists of four parts. In the first part, we describe the challenge of the pervasive and permanent philosophical disagreement over philosophers’ epistemic self-esteem. In the second part, we investigate the attitude of philosophers who have high epistemic self-esteem even in the face of philosophical disagreement and who believe they have well-grounded philosophical knowledge. In the third section, we focus on the attitude of philosophers who maintain a moderate level of epistemic self-esteem because they do not attribute substantive philosophical knowledge to themselves but still believe that they have epistemic right to defend substantive philosophical beliefs. In the fourth section, we analyse the attitude of philosophers who have a low level of epistemic self-esteem in relation to substantive philosophical beliefs and make no attempt to defend those beliefs. We argue that when faced with philosophical disagreement philosophers either have to deny that the dissenting philosophers are their epistemic peers or have to admit that doing philosophy is less meaningful than it seemed before. In this second case, philosophical activity and performance should not contribute to the philosophers’ overall epistemic self-esteem to any significant extent.

scholarly journals Self-splitting Abelian groups

2001 ◽  
Vol 64 (1) ◽  
pp. 71-79 ◽  
Author(s):  
P. Schultz
Keyword(s):  
Abelian Group ◽  
Direct Product ◽  
Free Module ◽  
Research Report ◽  
Original Form ◽  
Original Research ◽  
Direct Sum ◽  
The University ◽  
Torsion Free ◽  
Made In

G is reduced torsion-free A belian group such that for every direct sum ⊕G of copies of G, Ext(⊕G, ⊕G) = 0 if and only if G is a free module over a rank 1 ring. For every direct product ΠG of copies of G, Ext(ΠG,ΠG) = 0 if and only if G is cotorsion.This paper began as a Research Report of the Department of Mathematics of the University of Western Australia in 1988, and circulated among members of the Abelian group community. However, it was never submitted for publication. The results have been cited, widely, and since copies of the original research report are no longer available, the paper is presented here in its original form in Sections 1 to 5. In Section 6, I survey the progress that has been made in the topic since 1988.

The Concept of Political Equality: A Post-Dahl Analysis

1981 ◽  
Vol 29 (3) ◽  
pp. 392-406 ◽  
Author(s):  
Alan Ware
Keyword(s):  
Liberal Theory ◽  
Political Equality ◽  
Fourth Section ◽  
The Third ◽  
Introductory Section ◽  
Conservative Nature ◽  
Three Components

This article presents an analysis of the concept of political equality that is derived partly from the analysis of Robert Dahl. Following an introductory section, the conservative nature of Dahl's idea of political equality is outlined from four perspectives. With the last of these perspectives, a distinction between ‘populist’ and ‘liberal’ theories of democracy is introduced. In the third section it is argued that there are three components of political equality within a liberal theory, and the fourth section is devoted to an analysis of one of these—equality of treatment in the promotion of interests.

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