A Walk Through Weak Hyperstructures: Hv-Structures

347 Pages·2018·3.479 MB·English

by  Bijan Davvaz, Thomas Vougiouklis

A Walk Through
Weak
Hyperstructures
H -Structures
v

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A Walk Through
Weak
Hyperstructures
H -Structures
v
Bijan Davvaz
Yazd University, Iran
Thomas Vougiouklis
Democritus University of Thrace, Greece
World Scientific
NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI • TOKYO
11229_9789813278868_tp.indd 2 15/11/18 12:09 PM
Published by
World Scientific Publishing Co. Pte. Ltd.
5 Toh Tuck Link, Singapore 596224
USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601
UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
Library of Congress Cataloging-in-Publication Data
Names: Davvaz, Bijan, author. | Vougiouklis, Thomas, author.
Title: A walk through weak hyperstructures : H-structures / by Bijan Davvaz
v
(Yazd University, Iran), Thomas Vougiouklis (Democritus University of Thrace, Greece).
Description: New Jersey : World Scientific, 2019. | Includes bibliographical
references and index.
Identifiers: LCCN 2018048126 | ISBN 9789813278868 (hardcover : alk. paper)
Subjects: LCSH: Hypergroups. | Group theory. | Ordered algebraic structures.
Classification: LCC QA174.2 .D3845 2019 | DDC 511.3/3--dc23
LC record available at https://lccn.loc.gov/2018048126
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
Copyright © 2019 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,
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Preface
Hyperstructures were born as a generalization of an operation by the hy-
peroperation, from the single-valued operation to the multi-valued one.
It was then that the problem of generalizations was transferred into the
generalizations of axioms. In 1934 Frederick Marty, who introduced the
hyperoperation and gave the definition of the hypergroup, used the ‘dou-
ble’ axiom of reproductivity instead of the two axioms: the existence of
the unit element and the existence of the inverses. This is a revolutionary
generalization since the majority of the hyperstructures do not have unit
elements. Inmathematics, anygeneralizationofastructureshouldcontain
the generalized one as a sub-case. In hyperstructures the problem becomes
complicated as in the result we replace the elements by sets, in fact, we
replace a set by a power set. Therefore, we need new tools to achieve the
connection of the hyperstructures with the classical structures. This new
tool is the fundamental relation of each new hyperstructure. It is a fact
that any fundamental relation is based on the ‘result’. For example, in
hypergroups the fundamental relation β∗ is the transitive closure of the
relation β, where two elements are in β-relation if they belong to a hyper-
productoftwoelements. Inthefundamentalrelationβ∗,introducedbyM.
Koskas in 1970, a classical group corresponds to any hypergroup. In other
words, any hypergroup hides a group. The largest generalization in order
to have this correspondence, the existence of the β∗ fundamental relation,
is the one by using the so called weak axioms. In the weak axioms, defined
in all known classical structures as introduced by Vougiouklis in 1990, the
‘equality’ in any relation is replaced by the ‘non-empty intersection’ and
this leads to the largest class of hyperstructures called H -structures. The
v
mainwaytoprovetheoremsinthistopicisthereductiontoabsurdity. Since
the weak generalization is the most general, the number of H -structure is
v
v
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vi A Walk Through Weak Hyperstructures: Hv-Structures
dramatically big. Therefore, many problems in life and in other sciences
canbeexpressedbymodelsusingH -structures. Inordertospecifytheap-
v
propriate H -structure in models, one can use more restrictions or axioms
v
to reduce the number of possible cases. In generalizations new concepts
appear. Moreover, new axioms, new properties and new classes of hy-
perstructures, are discovered. Consequently, new classifications are needed
andveryinterestingmathematicalproblemsarerevealed. Thepresentbook
consists of seven chapters. Chapter 1 contains a fairly detailed discussion
of the basic ideas underlying the fundamentals of algebraic structures such
as semigroups, groups, rings, modules and vector spaces. Chapter 2 gives
a brief introduction to algebraic hyperstructures to be used in the next
chapters. Many readers, already familiar with these theories, may wish to
skip them or to regard them as a survey. In Chapter 3, the concept of
H -semigroups, H -groups and some examples are presented. Fundamen-
v v
talrelationsonH -groupsarediscussed. ReversibleH -groups, asequence
v v
of finite H -groups, fuzzy H -groups and H -semigroups as noise problem
v v v
are studied. In Chapter 4, we present the notion of H -rings. H -rings are
v v
the largest class of algebraic hyperstructures that satisfy ring-like axioms.
Weconsiderthefundamentalrelationγ∗ definedonH -ringsandgivesome
v
properties of this important relation. The fundamental relation on an H -
v
ringisthesmallestequivalencerelationsuchthatthequotientwouldbethe
(fundamental) ring. Then, we present several kinds of H -rings. In partic-
v
ular, we investigate multiplicative H -rings, H -fields, P-hyperoperations,
v v
∂-hyperoperations,H -ringoffractions,H -nearringsandfuzzyH -ideals.
v v v
Chapter 5 begins with the definition of H -module. Then the concepts of
v

H -module of fractions, direct system and direct limit of H -modules are
v v
provided. It is proved that direct limit always exists in the category of
H -modules. We study M[−] and −[M] functors and investigate the ex-
v
actness and some related concepts. Next, we prove Five Short Lemma and
Shanuel’slemmainH -modules. Attheendofthischapter,theconceptsof
v
fuzzy and intuitionistic fuzzy H -submodules are presented. In Chapter 6,
v
we cover H -vector space, hyperalgebra, e-hyperstructures and H -matrix
v v
representations. Moreover, we study Lie-Santilli theory. In the quiver of
hyperstructures Santilli, in early 90’es, tried to find algebraic structures
in order to express his pioneer Lie-Santilli’s Theory. Santilli’s theory on
‘isotopies’ and ‘genotopies’, born in 1960’s, desperately needs ‘units e’ on
left or right, which are nowhere singular, symmetric, real-valued, positive-
defined for n-dimensional matrices based on the so called isofields. These
elements can be found in hyperstructure theory, especially in H -structure
v
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Preface vii
theory introduced. This connection appeared first in 1996 and actually
several H -fields, the e-hyperfields, can be used as isofields or genofields,
v
in such way that they should cover additional properties and satisfy more
restrictions. Several large classes of hyperstructures as the P-hyperfields,
can be used in Lie-Santilli’s theory when multivalued problems appeared,
eitherinfiniteorininfinitecase. Chapter7,whichisnovelinabookofthis
kind, illustrates the use of weak hyperstructures. We present examples of
weak hyperstructures associated with chain reactions and dismutation re-
actions. ForthefirsttimeDavvazandDehghan-Nezhadprovidedexamples
ofhyperstructuresassociatedwithchainreactions. Also,weinvestigatethe
examples of hyperstructures and weak hyperstructures associated with re-
dox reactions and electrochemical cells. Another motivation for the study
of hyperstructures comes from biology, more specifically from Mendel, the
father of genetics, who took the first steps in defining “contrasting charac-
ters, genotypes in F and F ... and setting different laws”. The genotypes
1 2
of F are dependent on the type of its parents genotype and it follows
2
certain rules. Also, inheritance issue based on genetic information is ex-
amined carefully via a new hyperalgebraic approach. Several examples are
providedfromdifferentbiologypointsofview,andweshowthatthetheory
ofhyperstructuresexactlyfitstheinheritanceissue. Moreover,weprovidea
physicalexampleofhyperstructuresassociatedwiththeelementaryparticle
physics, the leptons. We consider this important group of the elementary
particles and show that this set along with the interactions between its
members can be described by the algebraic hyperstructures.

Bijan Davvaz
Department of Mathematics, Yazd University,
Yazd, Iran
Thomas Vougiouklis
School of Science of Education, Democritus University of Thrace,
Alexandroupolis, Greece
b2530 International Strategic Relations and China’s National Security: World at the Crossroads
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Contents
Preface v
1. Fundamentals of algebraic structures 1
1.1 Semigroups and groups . . . . . . . . . . . . . . . . . . . 1
1.2 Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.3 Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
1.4 Vector space. . . . . . . . . . . . . . . . . . . . . . . . . . 40
2. Algebraic hyperstructures 43
2.1 Semihypergroup . . . . . . . . . . . . . . . . . . . . . . . 43
2.2 Hypergroups . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.3 Hyperrings . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2.4 Hypermodules. . . . . . . . . . . . . . . . . . . . . . . . . 61
3. H -groups 69
v
3.1 H -groups and some examples . . . . . . . . . . . . . . . 69
v
3.2 Enumeration of H -groups . . . . . . . . . . . . . . . . . . 71
v
3.3 Fundamental relation on H -groups . . . . . . . . . . . . 76
v
3.4 Reversible H -groups . . . . . . . . . . . . . . . . . . . . . 81
v
3.5 A sequence of finite H -groups . . . . . . . . . . . . . . . 86
v
3.6 Fuzzy H -groups . . . . . . . . . . . . . . . . . . . . . . . 94
v
3.7 H -semigroups and noise problem. . . . . . . . . . . . . . 106
v
4. H -rings 115
v
4.1 H -rings and some examples . . . . . . . . . . . . . . . . 115
v
4.2 Fundamental relations on H -rings . . . . . . . . . . . . . 123
v
ix