scholarly journals Isomorphism of the characteristic semi- group of the direct sum and direct product of the asynchronous automatons of the strongly connected and determined analogs of their extensions

2018 ◽  
Vol 19 (12) ◽  
pp. 312-315
Author(s):  
Stanisław Bocian
Keyword(s):  
Direct Product ◽  
Semi Group ◽  
Parallel Calculation ◽  
Direct Sum ◽  
Strongly Connected ◽  
Abstract Class ◽  
Suitable Characteristic

In this article it is presented that the characteristic semi – group of the direct sum and direct product “G” and “AG” of the asynchronous automatons of the strongly connected and determined analogs of their extensions are isomorphism. Taking into account that the characteristic semi – group determines the ability to process the information then the direct sum and direct product can be consider as realization – the sequence and parallel calculation accordingly. The obtained results mean that this ability doesn’t depend on the sequence and parallel realization (the same number of abstract class of the suitable characteristic semi- groups).

scholarly journals Complexity of the characteristic semi-groups the direct products of „AG“ asynchronous automata of the strongly connected determined analogs the extensions associated with DFASC2 isomorfisms

2018 ◽  
Vol 19 (6) ◽  
pp. 1007-1011
Author(s):  
Stanisław Bocian
Keyword(s):  
Mathematical Model ◽  
Direct Product ◽  
Direct Products ◽  
Semi Group ◽  
Product Complexity ◽  
Asynchronous Automata ◽  
Strongly Connected ◽  
The Many ◽  
The Mathematical Model ◽  
Computational Processes

The paper presents the assumption and the evidence is carried out of the direct product complexity of character-istic semi-groups of any number (“ ”) of deterministic, finite, asynchronous, highly consistent DFASC2. automata. The characteristic semi-group of the automaton interferes in the computational algorithm of the generalized homoeo-morphism of the automatons. Then determination the com-plexity of the characteristic semi-group enables to estimate the complexity of the computational generalized homoeo-morphism for the other classes of automatons. In the range of the mathematical model the conception of the determined analog of the extension of the automaton associated with the isomorphism g0, g1 ,…, gq-1 where q is the grade of the extensions, with the suitable assumptions it simulates the automaton variable in time. The variable automaton in time is the adequate mathematical model for the many technical and computational processes of the real time. The direct product of automatons can be considered as the realization- parallel calculations accordingly

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.

Sylow Theory for a Certain Class of Operator Groups

1961 ◽  
Vol 13 ◽  
pp. 192-200 ◽  
Author(s):  
Christine W. Ayoub
Keyword(s):  
Direct Product ◽  
Finite Groups ◽  
Complete Lattice ◽  
Classical Group ◽  
Additive Group ◽  
Sylow Subgroups ◽  
Direct Sum

In this paper we consider again the group-theoretic configuration studied in (1) and (2). Let G be an additive group (not necessarily abelian), let M be a system of operators for G, and let ϕ be a family of admissible subgroups which form a complete lattice relative to intersection and compositum. Under these circumstances we call G an M — ϕ group. In (1) we studied the normal chains for an M — ϕ group and the relation between certain normal chains. In (2) we considered the possibility of representing an M — ϕ group as the direct sum of certain of its subgroups, and proved that with suitable restrictions on the M — ϕ group the analogue of the following theorem for finite groups holds: A group is the direct product of its Sylow subgroups if and only if it is nilpotent. Here we show that under suitable hypotheses (hypotheses (I), (II), and (III) stated at the beginning of §3) it is possible to generalize to M — ϕ groups many of the Sylow theorems of classical group theorem.

scholarly journals On Study of Vertex Labeling of Graph Operations

2011 ◽  
Vol 3 (2) ◽  
pp. 291-301
Author(s):  
M. A. Rajan ◽  
N. M. Kembhavimath ◽  
V. Lokesha
Keyword(s):  
Direct Product ◽  
Graph Labeling ◽  
Natural Numbers ◽  
Graph Vertex ◽  
Direct Sum ◽  
Graph Operations ◽  
Subdivision Graph ◽  
Maximal Vertex ◽  
The Given

Vertices of the graphs are labeled from the set of natural numbers from 1 to the order of the given graph. Vertex adjacency label set (AVLS) is the set of ordered pair of vertices and its corresponding label of the graph. A notion of vertex adjacency label number (VALN) is introduced in this paper. For each VLS, VLN of graph is the sum of labels of all the adjacent pairs of the vertices of the graph. is the maximum number among all the VALNs of the  different labeling of the graph and the corresponding VALS is defined as maximal vertex  adjacency label set . In this paper  for different graph operations are discussed. Keywords: Subdivision; Graph labeling; Direct sum; Direct product.© 2011 JSR Publications. ISSN: 2070-0237 (Print); 2070-0245 (Online). All rights reserved.doi:10.3329/jsr.v3i26222                  J. Sci. Res. 3 (2), 291-301 (2011) 

Direct Sums of Torsion-Free Covers

1973 ◽  
Vol 25 (5) ◽  
pp. 1002-1005
Author(s):  
Thomas Cheatham
Keyword(s):  
Direct Product ◽  
Commutative Rings ◽  
Integral Domains ◽  
Direct Sums ◽  
Artinian Rings ◽  
Direct Sum ◽  
Special Case ◽  
Torsion Free

In [4, Theorem 4.1, p. 45], Enochs characterizes the integral domains with the property that the direct product of any family of torsion-free covers is a torsion-free cover. In a setting which includes integral domains as a special case, we consider the corresponding question for direct sums. We use the notion of torsion introduced by Goldie [5]. Among commutative rings, we show that the property “any direct sum of torsion-free covers is a torsion-free cover“ characterizes the semi-simple Artinian rings.

scholarly journals On semigroups and groups of local polynomial functions

1979 ◽  
Vol 28 (2) ◽  
pp. 214-218
Author(s):  
Wilfried Nöbauer
Keyword(s):  
Positive Integer ◽  
Wreath Product ◽  
Direct Product ◽  
Polynomial Function ◽  
Subject Classification ◽  
Polynomial Functions ◽  
Semi Group ◽  
Factor Ring ◽  
Local Polynomial

AbstractLet Zn be the factor ring of the integers mod n and t be a positive integer. In this paper some results are given on the structure of the semigroup of all mappings from Zn into Zn and on the structure of the group of all permutations on Zn, which, for any t elements, can be represented by a polynomial function. If n = ab and a, b are relatively prime, then this (semi)group is isomorphic to the direct product of the respective (semi)groups for a and b. Thus it is sufficient to consider only the case where n = pe, p being a prime. In this case it is proved, that the (semi)group is isomorphic to the wreath product of a certain sub(semi)group of the symmetric (semi)group on Zpe−1 by the symmetric (semi)group on Zp. Some remarks on these sub(semi)groups are given.Subject classification (Amer. Math. Soc. (MOS) 1970): 20 B 99, 13 B 25.

scholarly journals On Arc Connectivity of Direct-Product Digraphs

2012 ◽  
Vol 2012 ◽  
pp. 1-6
Author(s):  
Tiedan Zhu ◽  
Jianping Ou
Keyword(s):  
Explicit Expression ◽  
Direct Product ◽  
Necessary Conditions ◽  
Strongly Connected ◽  
Sufficient And Necessary Conditions

Four natural orientations of the direct product of two digraphs are introduced in this paper. Sufficient and necessary conditions for these orientations to be strongly connected are presented, as well as an explicit expression of the arc connectivity of a class of direct-product digraphs.

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.

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