Smarandache special elements in multiset semigroups

2018·1.7 MB·English

by  Vasantha Kandasamy, W. B.

Smarandache Special
Elements in Multiset
Semigroups

W. B. Vasantha Kandasamy
Ilanthenral K

2018
Copyright 2018 by EuropaNova ASBL and the Authors
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Peer Reviewers:
Said Broumi, University of Hassan II Mohammedia, Hay El Baraka Ben
M'sik, Casablanca B. P. 7951, Morocco.
Prof. Ion Patrascu, Department of Mathematics, Fratii Buzesti National
College, Craiova, Romania.
Dr. Azeddine Elhassouny, Département de Génie logiciel ENSIAS,
Avenue Mohammed Ben Abdallah Regragui, Madinat Al Irfane, BP 713
Agdal Rabat, Morocco.
Many books can be downloaded from the following
Digital Library of Science:
http://fs.unm.edu/ScienceLibrary.htm
ISBN-13: 978-1-59973-564-1
EAN: 9781599735641
Printed in EU
2
CONTENTS

Preface 4

Chapter One
BASIC CONCEPTS 7

Chapter Two
SMARANDACHE ELEMENTS IN
MULTISET SEMIGROUPS 17

Chapter Three
N-MULTIPLICITY– MULTISETS AND
THEIR ALGEBRAIC PROPERTIES 53

3
Chapter Four
MULTISET MATRICES 175

FURTHER READING 237

INDEX 241

ABOUT THE AUTHORS 243

4
PREFACE

Authors in this book study the notion of Smarandache
element in multiset semigroups. It is important to keep on
record that we define four operations on multisets viz. +, X,
union and intersection in a free way. Thus all sets finite or
infinite order contribute to infinite order multisets and the
semigroup under any of these operations is of infinite order.
We in this book define a new notion called n- multiplicity
multiset using any set S, denoted by n-M(S). This n-multiplicity
multiset contains all set got using S where the number of times
an element in the multiset M(S) cannot repeat more than n-
times. On n-multiplicity multisets we cannot define + or  or 
as in case of multisets, for closure axiom fails in all cases. We
overcome this problem by the method of levelling. We have
defined l(+) where l denotes the levelling of addition or levelled
addition. It is proved that n-multiplicity of M(S) under l(+) is a
semigroup of finite order if S is finite and if S is infinite {n-
5
multiplicity M(S) , l(+)} is a commutative semigroup of infinite
order. Similarly in case of  and  we use levelling.
Several interesting properties are derived and one of the
innovations made here is the defining special Smarandache
elements on these semigroups, like Smarandache special
idempotents, Smarandache special zero divisors and so on.
We wish to acknowledge Dr. K Kandasamy for his
sustained support and encouragement in the writing of this
book.

W.B.VASANTHA KANDASAMY
ILANTHENRAL K

6
Chapter One

BASIC CONCEPTS

In this chapter we just proceed onto define some basic
concepts which are essential for an easy understanding of the
other chapters and the algebraic structures defined on multisets.
We first list out the properties of power set, semilattices and
lattices.
A power set of a nonempty set S is the collection of all
subsets of S together with S and  denoted by P(S); P(S) = {All
subsets of S including S and the empty set }.
Example 1.1. Let S = {a , a , a , a } be set of four elements the
1 2 3 4
power set of S is P(S). P({a , a , a , a }) = {{a }, {a }, {a },
1 2 3 4 1 2 3
{a }, , {a , a }, {a , a } {a , a }, {a , a } {a , a }, {a , a }, {a ,
4 1 2 1 3 1 4 2 3 2 4 3 4 1
a , a }, {a , a , a }, {a , a , a }, {a , a , a }, {a , a , a , a }}.
2 3 1 2 4 1 3 4 2 3 4 1 2 3 4
Clearly |P(S)| = 24 so if |S| = n then |P(S)| = 2n and the power set
of a finite set is always finite.
It is well know P(S) under union of set ‘’ is a
semilattice.
8 Smarandache Special Elements in Multiset Semigroups

Further P(S) is a partially ordered set {P(S), } is a
commutative idempotent semigroup of finite order.

Similarly P(S) under intersection of set is a semilattice
or {(P(S), } is again a finite idempotent semigroup which is
commutative.
Further it has been proved that {P(S), , } is a lattice
and this lattice is complemented and distributive hence a
Boolean algebra of order 2|S|. We would be using this concept as
the multisets in general under the unrestricted union and
intersection is not a Boolean algebra. Further the order of
multisets of any set S finite or infinite is of infinite order.
For instance if S = {a , a , a } we see P(S) = {, {a },
1 2 3 1
{a }, {a }, {a , a }, {a , a }, {a , a }, {a , a , a }} is the power
2 3 1 2 1 3 2 3 1 2 3
set associated with S and |S| = 8 = 23. The Boolean algebra B
associated with P(S) is as follows.
{a , a , a }
1 2 3

{a , a }
B = 1 2 {a , a }
{a , a } 2 3
1 3

{a }
1 {a }
2 {a }
3

{}
Figure 1.1
We define B as a Boolean algebra of order 8.
Basic Concepts 9
Now we proceed onto describe multisets and the
operations which we have used in this book.
Definition 1.1: Let S be a finite or infinite set. Multiset of S
denoted by M(S) = {collection of all subsets of S where in the
subsets any element of S is repeated from one time to infinitely
many times)}.
Thus if S ={a}; a singleton set, the multiset of S denoted
by M(S) = {, {a}, {a, a}, {a, a, a} … , {a, a, a, ….-infinite
number of times}. Hence M(S) is always infinite immaterial of
whether S is finite or infinite.
(i). In this first place a multiset of any set S finite or infinite is
always infinite and is denoted by M(S).
(ii) We illustrate how  operation is defined on M(S).
If A and B  M(S), A  B contains all elements of A and
all elements of B; no restrictions on the repetitions of the
elements.
Thus if A = {a , a , a , a , a , a , a , a , a } and B = {a , a ,
1 1 1 2 2 3 8 8 8 1 1
a , a , a , a , a , a , a , a , a }  M(S) where S = {a , a , …, a }
2 2 2 2 2 2 2 9 9 1 2 9
A  B = {a , a , a , a , a , a , a , a , a , a , a , a , a , a , a , a , a ,
1 1 1 2 2 3 8 8 8 1 1 2 2 2 2 2 2
a , a , a , a }.
2 2 9 9
This is the way we have define ‘’ operation on
multisets.
Hence if we take infinite union of many sets then the
resulting set is infinite though |S| = 9.