Pseudo Lattice Graphs and their Applications to Fuzzy and Neutrosophic Models

W. B. Vasantha Kandasamy Florentin Smarandache Ilanthenral K

2014 This book can be ordered from:

EuropaNova ASBL Clos du Parnasse, 3E 1000, Bruxelles Belgium E-mail: [email protected] URL: http://www.europanova.be/

Copyright 2014 by EuropaNova ASBL and the Authors

Peer reviewers:

Dr. Stefan Vladutescu, University of Craiova, Romania. Dr. Octavian Cira, Aurel Vlaicu University of Arad, Romania. Mumtaz Ali, Department of Mathematics, Quaid-i-Azam University, Islamabad, 44000, Pakistan Said Broumi, University of Hassan II Mohammedia, Hay El Baraka Ben M'sik, Casablanca B. P. 7951. Morocco.

Many books can be downloaded from the following Digital Library of Science: http://www.gallup.unm.edu/eBooks-otherformats.htm

ISBN-13: 978-1-59973-296-1 EAN: 9781599732961

Printed in Europe and United States of America

2 CONTENTS

Preface 5

Chapter One INTRODUCTION 7

Chapter Two PSEUDO LATTICE GRAPHS OF TYPE I 9

Chapter Three PSEUDO LATTICE GRAPHS OF TYPE II 99

Chapter Four PSEUDO NEUTROSOPHIC LATTICE GRAPHS OF TYPE I AND TYPE II 179

3 Chapter Five SUGGESTED PROBLEMS 223

FURTHER READING 261

INDEX 274

ABOUT THE AUTHORS 276

4

PREFACE

In this book for the first time authors introduce the concept of merged lattice, which gives a lattice or a graph. The resultant lattice or graph is defined as the pseudo lattice graph of type I. Here we also merge a graph with a lattice or two or more graphs which call as the pseudo lattice graph of type II. We merge either edges or vertices or both of a lattice and a graph or a lattice and a lattice or graph with itself.

Such study is innovative and these mergings are adopted on all fuzzy and neutrosophic models which work on graphs. The fuzzy models which work on graphs are FCMs, NCMs, FRMs, NRMs, NREs and FREs. This technique of merging FCMs or other fuzzy models is very advantageous for they save time and economy. Moreover each and every expert who works on the problems is given equal importance.

5 We called these newly built models as merged FCMs, merged NCMs, merged FRMs, merged NRMs, merged FREs and merged NREs.

We wish to acknowledge Dr. K Kandasamy for his sustained support and encouragement in the writing of this book.

W.B.VASANTHA KANDASAMY FLORENTIN SMARANDACHE ILANTHENRAL K

6

Chapter One

INTRODUCTON

In this chapter we just give some of the properties enjoyed by graphs and lattices. For in this book we obtain new classes of lattice-graphs by merging two lattices or by merging a lattice and a graph or a graph and a graph.

We use the term merging as follows. We may merge a vertex of a lattice L1 with another vertex or L2 or a edge and two vertices of a lattice L1 with an edge and two vertices of a lattice L2 or merge many vertices and many edges of a lattice L1 with that of a lattice L2.

Such study is new and innovative.

It goes without saying that every lattice is a connected graph but a graph in general is not a lattice, for

v2 v1  

 v3

v5 

 v 4 8 Pseudo Lattice Graphs and their Applications to Fuzzy…

is a graph and not a lattice. Further

v1 v8    v2

v9   v5  v7  v3

v10   v  4 v6

is a graph and not a lattice.

Now when in a lattice L1 merged by a vertex or edge or both with another lattice L2 we get the resultant graph which is termed as a pseudo lattice graph of type I it may be a lattice or a graph. Similarly using a lattice and a graph or a graph and a graph we get a graph termed as the pseudo lattice graph of type II.

This notion finds its applications in fuzzy and neutrosophic models which work on direct graphs like FCMs, NCMs, FRMs, NRMs, FREs and NREs [79, 90].

This book also studies about merging of neutrosophic lattices [87]. Thus these new type of merging may also find more applications in due course of time. Several open problems are suggested.

8

Chapter Two

PSEUDO LATTICE GRAPHS OF TYPE I

In this chapter we introduce a new mode of construction of graphs using lattices. We take two lattices merge one vertex or two vertices or three vertices or so on or merge one edge and two vertices or more edges and more vertices and arrive at a diagram. The resultant can be a graph or a lattice.

We will first illustrate all these situations by some examples.

Example 2.1: Let L be the chain lattice C 1 7 1 

 a1

 a 2

 a3

 a4

 a 5  0 10 Pseudo Lattice Graphs and their Applications to Fuzzy…

1 and L2 = 

x1   x2

 0

be the distributive lattice of order four.

We have the following ways of merging L1 and L2 and are denoted in the following.

1= 1 

x1   x2

a = 0  1

 a2 = M1  a3  a 4

 a 5  0

Merging vertex 1 of L1 with vertex 1 of L2 and zero of L2 with a1 of L1, we get the lattice given above. We can rename the vertices. Pseudo Lattice Graphs of Type I 11

1 

x1   x2

 0=1

 a1 = M2  a2

 a3

 a4

 a 5  0

Here vertex 1 of L1 is merged with 0 vertex of L2 and is denoted M2.

1 

 a1 = 1

 a2 x1   x2 = M3  a3

 a4= 0

 a5

 0

This sort of merging can be made which is self explanatory and the lattice is denoted by M3. 12 Pseudo Lattice Graphs and their Applications to Fuzzy…

1 

a1 

 a 2

 a3 and 1

= M4 x1   a4  x2

 a and 0 5  0

denoted by M4.

Let us merge in the following way 1 of L1 is merged with 1 of L2 and 0 of L1 is merged with 0 of L2.

1   a1

 a2 a x1   3  x2 = M5

 a4

 a5

 0 and 0

This merging is denoted by M5.

Pseudo Lattice Graphs of Type I 13

This merging is denoted by M6.

1 

 a1

 a2

 a3

x1   x2  a4 = M6

 a5  0

This merging is denoted by M7 .

1   1 and a1

 a2

 a3 x1   x2 = M7  a4

 a5  0

14 Pseudo Lattice Graphs and their Applications to Fuzzy…

Consider the merging of the vertices

1 

x 1   x2 = 1

 a1  0  a2

 a3

 a4

 a5  0 This is not a lattice only a graph.

Now consider the merging of 0 with x2 which is as follows:  1

 a1

 a2

 a3 1   a4

0  = x2  x1

 0

This is also a pseudo lattice graph which is not a lattice. Pseudo Lattice Graphs of Type I 15

Now we merge a2 with x1 which is as follows. The resultant is not a lattice only a graph.

 1 1

  a 1

x2   x = a 1 2

 a3  0  a4

 a5

 0 We can merge a1 with x1 or a2 with x1 or a3 with x1 or a4 and in all cases we get only a graph and not a lattice.

We see merging 1 with 1 and a2 with zero and we get a lattice which is modular. This is as follows:

1 

x1   a1  x2

 a2=0

 a3

 a 4

 a5  0

Clearly the above figure is a lattice and is a modular lattice.

Suppose x1 is merged with 1 and x2 is merged with a1 we get the following graph. 16 Pseudo Lattice Graphs and their Applications to Fuzzy…

1 

a 2 a4 x1 = 1        x2 = a1 a3 a5 0

 0

Clearly this is not a lattice.

1 

a2 a4 1        x1 = a1 a3= x2 a5 0  0

We can also merge x1 with a1 and x2 with a3 and get a graph which is not a lattice. We can merge a2 with x1 and a4 with x2 which is also follow:

1 

a = x 2 1 a4 = x2 1        a1 a3 a5 0

 0

This is also only a graph and is not a lattice.

Finally we can merge 0 with x2 and 1 with x1 which is as follows: Pseudo Lattice Graphs of Type I 17

1 

a 2 a4 1        a1 a3 a5 0

 0

Clearly the resultant is graph and not a lattice.

 1

 a 1

 a2

 a3

 a4

 a 5  0=1

x1   x2

 0

The merging of 0 of C6 with 1 of lattice we get the resultant is a lattice which is distributive.

We see when we merge two distributive lattice we can get the resultant as a distributive lattice or a modular lattice or only a graph which is not a lattice.

We can also merge x2 with 1 and get the following graph. 18 Pseudo Lattice Graphs and their Applications to Fuzzy…

1 

a2 a4 x 1        1=x2 a1 a3 a5 0  0

or merge 0 with x1 we get the following graph.

1 

      x1=0  x2 1 a1 a2 a3 a4 a5

 0

We can also merge x1 with a1 which is as follows:

1  x1=a1         x2 0 a a a a 1 5 4 3 2  0

Clearly this is also a graph and is not a lattice.

We can merge x2 with a2 and get the following graph which is as follows. 1 

x2= a2 a4 x1        1 a1 a3 a5 0

 0 Pseudo Lattice Graphs of Type I 19

We see if we merge a4 with x2 we get the following graph

1 

a4 =x2 a5 x1       

1 a1 a2 a3 0

 0

We get if we merge 0 with x2 then we get the following graph. 1 

x1        1 a1 a2 a3 a4 a5 0=x2

 0

We also merge 1 with 1 horizontally.

1       a1 a2 a3 a4 a5 0

x1   x2

 0 20 Pseudo Lattice Graphs and their Applications to Fuzzy…

This is also only a graph and not a lattice.

We can merge a3 with 1 horizontally and get the following graph.

1 a5 a4 1=a3 a2 a1 0       

x1   x2

 0

We can also merge 0 with a4 and get the following graph.

1 

x1   x 2 0        a a2 a1 0 a5 0=a4 3

Now if we can merge one edge with another edge we get only a graph which is as follows:

1 

x2 = a1 x1    a2  a3 0 a4  a  5 0

Pseudo Lattice Graphs of Type I 21

We can merge 1 with a1 and x2 with a2 which is as follows. 1  x2 = a1

a = x x1   2 2

a3

a4   a 0  5 0

We can merge 1 with a4 and x2 with 0 which is the following graph.

1 

a1

 a2

a3  a  4 1=a5

x1  x2=0

 0

We can also merge x1 with 0 and 1 with a5 and the edge 1 x1 with 0a5 given by the following graph.

 1

 a1

 a2 a3 a4

1=a5

x1 =0  x2

 0 22 Pseudo Lattice Graphs and their Applications to Fuzzy…

We can also merge x2 with a2 and 1 = a1 and the edge 1x2 with a2a1 and get the related graph that is as follows:

 1

 1=a1

 x x2=a2 1 a3  a4 0 a5 0 

Now we can merge the edge 0x1 with edge a3a2 so that

 1 1  a1   x2 x1= a2

 a3 = 0  a  4 a5 0 

Clearly this is not a lattice only a graph.

 1  a 1  a2 1   a3  a4   x2 x1 = a5



0

Pseudo Lattice Graphs of Type I 23

The following observations are to be made while merging a vertex of two lattices or merging only two vertices and not an edge of two lattices or merging an edge and two vertices of the lattices. From the example one it is clear that when we had used lattices both of which are distributive we may get a distributive lattice or a modular lattice or a graph which is not a lattice.

However after giving one to two more examples we proceed onto define the concept mathematically.

Example 2.2: Let L be the lattice

1 

  a a1 2  0

We merge the vertices of L with L or edges and vertices of L with L.

Some of the merging are described in the following.

Merge vertex 0 with vertex 1 of L.

1 

  a2 a1

 0m1

a =a 1 3   a2 = a4

 0

This new graph is a lattice which is distributive. 24 Pseudo Lattice Graphs and their Applications to Fuzzy…

We merge two lattices we can rename the vertices 1m0 or 0m1 means zero is merged with one or equivalently one is merged with 0.

1 1  

  a2m.a1  a a1 2   0 0

a2 is merged with a1 the resultant is not a lattice only a graph. We can also merge a1 with 1 of L. The resulting graph is as follows.

1 

a with 1   1 a2 a1   a2 0  0

a2 merged with 1.

1 

a2=1  a1 a2  a1   0  0

Pseudo Lattice Graphs of Type I 25

is only a graph not a lattice.

Merging 1 with a1 1  a =1 1   a2 a  1 a2   0  0

Merging a1 with 0 of the lattices

1 

a2  a1   1

 a m 1 0  a2  0

Now we can merge one with one.

1 

 a1 a2  a1   a2

 0  0

is only a graph. 26 Pseudo Lattice Graphs and their Applications to Fuzzy…

1  1 

a1   a2  a1 a2

 0

Merging 0 with 0 of the lattice L with L.

1 

 1

a1   a2  a2 0

 0

Merging a2 with a2 we get the following graphs.

1 

 1

a2   a1  a1 0 

 0

Now we can also merge only two vertices of L and get the graphs.

Pseudo Lattice Graphs of Type I 27

Some of them are given in the following:

1 

a1   a1  a2  a 2

 0

The resulting diagram is a lattice which is modular.

1   1

a1 = a1   a2 = a2 0   0

The resulting diagram is only a graph and not a lattice.

Thus using a distributive lattice we may get after merging its vertices a distributive lattice or a modular lattice or a graph. Now suppose we merge three vertices and not their respective edges we can get the following graphs

a1 = a1 

1=1  0   0

 a2 = a2 28 Pseudo Lattice Graphs and their Applications to Fuzzy…

or a1   a1 1 = 1   0=0

 a2=a2

in both cases we get the same graph.

Now we can also get the graph by merging 3 of its vertices and two its edges.

1=1 

a2 = a2  a1   a1

 0=0

This is clearly a modular lattice. The other way of merging the vertices with the edges is as follows.

1   1 a2   a1

 0=0 Pseudo Lattice Graphs of Type I 29

1 

a2=a2   a1=a1

0   0

Example 2.3: Let L1 =

 a1

 a2

 a3

 a4

a5   a6  a7

be a lattice with a7 the least element and a1 the greatest element.

L2 =

b1  b 3  b2 b4  b5 is a pentagon lattice.

30 Pseudo Lattice Graphs and their Applications to Fuzzy…

We first merge the vertex a1 with b1 we have the following graphs.

b1=a1 

 a2 b3 

 a3

 b2  a4 b4  a5   a6

 a7

b 5

or

 a =b b 1 1 3  a2  b2  a3 b4   a4

b5 a5    a6

 a7

Pseudo Lattice Graphs of Type I 31

or a1 

b 3 a2

 b2 b4 

 a3

 b5

 a4

a5   a6

a 7

or a1=b1 

b3   b2

 a2 b4  b5  a3

 a4

a7   a6  a7 32 Pseudo Lattice Graphs and their Applications to Fuzzy…

b1 = a1 

b3   b2

b4   b5  a2

 a3

 a4

a5   a6

 a 7

In all these cases we see the resultant is only a graph and not a lattice.

We define a pseudo lattice graph of type I as the lattice or a graph got by merging two lattices by a vertex or vertices an edge or edges or both.

We see we have at least two lattices built using the lattices L1 and L2 and both of them are non distributive and non modular as they contain a pentagon lattice as a sublattice which is both non modular and non distributive.

Pseudo Lattice Graphs of Type I 33

We can also merge vertex b1 with a2 and obtain the following.

 a 1  b = a 1 2 b 3  a3  b2  a4

b4  a5   a6  a7  b 2

or  a 1

 b1 = a2 b3   a3  b2

b4

 a4

 a5   a6 b5  a7

We see both the graphs are not lattices.

They are a special type of graphs with same number of vertices and edges. 34 Pseudo Lattice Graphs and their Applications to Fuzzy…

 a1

 b1 = a2 b 3  a3  b 2 b4 

 b 5  a4

a5   a6

 a 7

 a 1

 a2 = b1 b3   b2 b4  b5 

 a3

 a4

a5   a6

 a7

are all pseudo lattice graphs of type I which are only graphs and not lattices.

On similar lines we can merge vertex b1 with a3 and we have the following graphs. Pseudo Lattice Graphs of Type I 35

 a1

 a2

 a3 = b1

b  a4 3

a5   a6  b 2 b4  a7  b5

or

 a 1  a2

 a = b 3 3 b 3

 b2 b4 

 b5  a4

a5   a6

a 7

are pseudo lattice graphs of type I which are only graphs and not lattices. 36 Pseudo Lattice Graphs and their Applications to Fuzzy…

By merging the vertex a4 with b1 we get the following four types of graphs.

 a1

 a2

 a3 = b1

 a4 b3 a5  a6   b2

b4   a7  b5

or

 a1

 a2

 a3

 b1 = a4 b 3

 b2 b4 a5   a6

 b5

a 7

Pseudo Lattice Graphs of Type I 37

or

 a1

a  2

 a3

b a = b 3  4 1

a5   a6

 b2

b4  a7

 b5

or  a 1  a2

 a3  a4=b1

b3 a 5   b2  a6

b4 b5 



a7

All of them are pseudo lattice graphs of type I which are not lattices only graphs.

38 Pseudo Lattice Graphs and their Applications to Fuzzy…

 a1

 a2

 a3  a4

a5   a6

b  3 a7= b1  b2 b4  b5

is a lattice which is not modular.

We can merge the vertex b1 with a5 or a6 and get pseudo lattice graphs of type I which are only graphs.

 a1

 a2

 a3

 a4

a6   a5 = b1

  b2 a7  b3 b4  b5

Pseudo Lattice Graphs of Type I 39

or

 a 1  a2

 a 3  a4 b1=a6  a  5

b3   b a7 2  b4   b5

are only graphs.

Now by merging the vertex a6 or a5 with b2 or b3 or b5 or b4 we get the following pseudo lattice graphs of type I.

 a1

 a2

 a3

b  a4  1

a5   a6 = b3

  b2 a7 b 4  or b5 40 Pseudo Lattice Graphs and their Applications to Fuzzy…

 a 1  a2 b1  a  3 b3   a4  b2 a5  a = b  6 4  a7  b5

or

 a1 b  1 a2  b 3 a3  b2 b4 a4 

a5   a6 = b5

 a7

is again a pseudo lattice graph of type I which is a graph and not a lattice.

Let us merge the vertex b2 with a5 we get

 a1

 a2

b  a  1 3 a b3   4

b2 = a5   a6

b4    a7 b5 Pseudo Lattice Graphs of Type I 41

a pseudo lattice graph of type I which is only a graph.

We can merge b4 with a5 and get the following pseudo lattice graph.

 a1

 a2

b1 a3  a  4

b3  b2   a6

b4    a7 b5

This is only a graph and not a lattice and so on.

Next merge only two vertices to get the pseudo lattice graph of type I.

a1 = b1  b2

b3 b4   a2 = b5 a3

a 4 

a5   a6

 a7

The above pseudo lattice graph of type I is a lattice which is non distributive and non modular. 42 Pseudo Lattice Graphs and their Applications to Fuzzy…

a1 = b1  b2 a2 b3 b4   a3 = b5

a4  a5   a6



a7

is a pseudo lattice graph of type I which is again a non distributive and non modular lattice.

Now

a1 = b1  b2

a2 b3 b4   a35

 a4 = b5

a5   a6  a7

is a pseudo lattice graph of type I which is a non distributive and non modular lattice.

Pseudo Lattice Graphs of Type I 43

Now

a1 = b1 

a2  b2  b3  a3

 a4 a 5  b4   a6

 a = b 7 5

is a pseudo lattice graph of type I.

Now consider

a1 

a  b1 2

 a b3  3

 a4

b4 = a6   b = a 2 5 a 7 



b5 44 Pseudo Lattice Graphs and their Applications to Fuzzy…

a1 

 a2

 a3 b1   a4

b2 = a5  b = a  3 6   b4 a7  b5

or

a1 

 a2

b 1   a3

 b3  a4

b2 = a6   a5= b4

 a7  b5

We can also have the merging of three vertices Pseudo Lattice Graphs of Type I 45

a1 

 a2

 a3

 b1 = a4

a6 = b3  b2 = a5

  b4 a7  b5 and so on.

We give a few illustration of merging edges as well as vertices

b 1

 b2 a1

 b3  a2   a 3 b5  a4

a5   a6

 a7 is only a graph not a lattice.

46 Pseudo Lattice Graphs and their Applications to Fuzzy…

We get a pseudo lattice graph by merging edges b3b5 with a6a7 which is as follows.

 a1  a2 b 1  a3 b2  a4   a b3 = a6 b  5  4  a3 a7 = b5

is not a lattice only a graph.

We can merge a5a7 with b4b5 and get the pseudo lattice graph which is as follows:

 a1

 a2

a3   b 1 b2 = a5  a 4  b3   a6 b4 = a7  b5

We can also merge edges b1b2 with a6a7 and get the following pseudo lattice graph of type I.

Pseudo Lattice Graphs of Type I 47

 a1

 a2

a5 a   3

a6  a4 = b1

 b2=a7

 b4

b3   b5

We can also merge edges a4a6 with b1b2 and get the following pseudo lattice graph which is only a graph and not a lattice.

 a1

 a2  a3

 a4 = b1

b2=a6  a5   b3  a7 b4   b 5

Likewise we can merge an edge and a vertex and get a pseudo lattice graph of type I.

We can merge 2 edges and three vertices and get the following few pseudo lattice graphs of type I. 48 Pseudo Lattice Graphs and their Applications to Fuzzy…

 a1

 a2

a  3

 a4 = b1

b2  a  b3 = a5  6

b4   b5 = a7

This is a lattice.

We can also merge edges b1b2 and b2b4 with a4a6 and a6a7 respectively.

 a1

 a2  a3

 a4 = b1

b2=a6   a5  b3

b4   b 5

This is also pseudo lattice graph of type I which is also a lattice.

We can also merge four vertices and three edges which is a pseudo lattice graph of type I which is as follows. Pseudo Lattice Graphs of Type I 49

 a1

 a2  a3  b1

b2=a4   b = a 3 5

b4 = a6   b5 = a7

We can merge three edges with four vertices in the following way the get the pseudo lattice graph of type I.

 a1

 a2

a  3

 a4 = b1

b2=a6   b = a 3 5

b4 = a7   b5

This is a lattice which is both non distributive and non modular.

We can also merge four vertices and three edges and obtain the following pseudo lattice graph of type I which is as follows.

50 Pseudo Lattice Graphs and their Applications to Fuzzy…

 a1

 a2

a  3

 a4

b2=a5   b = a 3 6

b4   b5 = a7

This is a lattice which is not modular and non distributive. We can also still differently obtain several other graphs.

Thus we propose the following open problems.

1. Given two lattices L1 and L2 how many pseudo lattice graphs of type I can be got (|L1| = n, |L2| = m that is number of vertices of L1 is n and that of L2 is m and number of edges of L1 is s and that of L2 is t)?

2. How many of the pseudo lattice graphs of type I are lattices?

3. If both L1 and L2 are distributive lattices, can the pseudo lattice graph which is a lattice be a modular lattice?

4. How many pseudo lattice graphs of type I that is obtained by merging one vertex is a lattice?

Pseudo Lattice Graphs of Type I 51

5. How many pseudo lattice graphs of type I can be obtained by merging two vertices but not an edge is a lattice?

6. How many pseudo lattice graphs of type I can be obtained by merging two vertices and an edge of lattices? Since this study is very new the six problems proposed above can be realized as open conjectures.

Example 2.4: Let L1 =

a1 

a2   a3  a4  a5

and L2 = b1 

b2   b3  b4

b6   b5  b7 be the two lattices.

Suppose we merge vertex b7 with vertex a1 then we get the following pseudo lattice graph of type I.

b1 

b b 2   b3  4

b6   b5  b7 = a1

a2   a3  a4

 a5 52 Pseudo Lattice Graphs and their Applications to Fuzzy…

If we merge b1 with a5 we get the following pseudo lattice graph of type I.

a1 

a2   a3  a4

 a5 = b1

b2   b3  b4

b6   b 5  b7

Suppose we merge the vertex b4 with a1 we get the following pseudo lattice graph of type I.

b1 a1   b4=a2 b2   b3   a3  a4

b6    b5 a5  b7

This is not a lattice only a graph.

By merging vertex b5 with a1 we get the following pseudo lattice graph of type I.

a1 b1   a3  a2  b2=a4 b   3  b4 a5   b6  b5  b7 Pseudo Lattice Graphs of Type I 53

This is only a graph and not a lattice.

By merging vertex b6 with vertex a4 we get the following pseudo lattice graph of type I.

b1  a 1 b  2   b3  b4

a3   a2   b6=a4 b5   a 5 b7

This is a graph and not a lattice. Let us merge b3 with a1;

b1 

b2 b3= a1    b4

a3 b6    a2 a4 b5    b7 a5

is the pseudo lattice graph of type I which is only a graph and not a lattice.

Merging of a5 with b3 gives the following pseudo lattice graph of type I.

a1  b1 a3   a2  a4

b2   b3  b 4 b6   b5  b7 54 Pseudo Lattice Graphs and their Applications to Fuzzy…

This is only a graph and not a lattice we can get several such pseudo lattice graphs of type I.

We can give a few pseudo lattice graphs of type I by merging only two vertices and not an edge.

b1 a1  

b2   b3   a4 a = b 2 4 b6=a4    a b5 5  b7

the vertex a3 and b3 and a2 and b4 are merged.

a1 b1  

a4  b2   b3  b4

b6=a4    a5 b5  b7

This is also a graph. So the pseudo lattice graph got by merging the vertex a3 with b2 and vertex b3 with a4 is only a graph and not a lattice.

b1 

b2=a1   b3  b4

b6  a3  a2  b5= a4  b =a 7 5 Pseudo Lattice Graphs of Type I 55

The pseudo lattice graph got by merging the vertices b2 with a1, a 4 with b5 and b7 with a5 we get the graph which is not a lattice.

We can merge four vertices and get the following pseudo lattice graph.

b1 = a1  b2=a3  a2  b4 = a4

b0   b5 = a5  b7

This pseudo lattice graph is obtained by merging the vertices a1 with b1, a4 with b4, a5 with b5 and edge b1 b4 with edge a1a4 and edge b4b5 with edge a4a5. This pseudo lattice graph of type I is a lattice which is modular.

b1=a1  b2=a2    b4=a4 b3=a3 b6   b5 = a5 

b7

Thus using these two modular lattices we get pseudo lattice graph of type I which is a modular lattice in some cases and just only graphs in many cases.

We see by merging the vertices a1 with b1, a2 with b2, a3 with b3, a4 with b4 and a5 with b5 and edges a1a2 with b1b2 a1a3 with b1b3, a1a4 with b1b4; b2b5 with a2a5 and b3b5 with a3a5 we get the pseudo lattice graph of type I is the lattice L2.

56 Pseudo Lattice Graphs and their Applications to Fuzzy…

We can also merge in this form

b1 

a1 = b2   b3  b4 a3 = b6   a2  b5 = a4  b = a 7 5

to get a pseudo lattice graph of type I where the vertex a1 is merged with the vertex b2, vertex a is merged with vertex b6, vertex b5 is merged with vertex a4 and vertex b7 is merged with vertex a5, the corresponding edges are also merged. The resultant, pseudo lattice graph of type I is a modular lattice.

Example 2.5: Let L1 = a1  a2 and b1  a3  b2  a4  b3  a5 b  4  a6

be two chain lattices.

Only in one case when vertex a1 is merged with vertex b4 or vertex a6 is merged with vertex b1 we get the pseudo lattice graph of type I to be a chain lattices in all other the pseudo lattice graph of type I is only a graph more so it is a tree.

A few merging of vertices and edges of L1 with L2 is given in the following. Pseudo Lattice Graphs of Type I 57

a1 = b1 a2   b2

a3   b3 a4   b4 a5  a6 

is a tree.

b 1  b2=a1 a2   b3 a3   b4 a4

a5  a6 

The pseudo lattice graph of type I is a tree got by merging vertex a1 with vertex b2.

a1 a  2 b1

b  a3=b2 3 b  a4 4  a5

 a 6

The pseudo lattice graph of type I is a tree got by merging vertex a3 with vertex b2.

58 Pseudo Lattice Graphs and their Applications to Fuzzy…

a1

 a2

 a3 a =b b  4 1 2  b3  a5  b4   a6

The pseudo lattice graph of type I got by merging vertex a4 with vertex b1.

 a1 b 1  a2  b2 b3 = a3 

  a4

b4  a5

 a6

is a pseudo lattice graph of type I got by merging vertex a3 with b3.

We can merge a maximum of 3 edges and 3 vertices in which case we get a chain lattice.

Pseudo Lattice Graphs of Type I 59

b1  a 1  a2=b2  a3=b3

 a4=b4

 a5

 a6

is a true which is a pseudo lattice graph of type I got by merging the vertices a2 with b2, a3 with b3 and a4 with b4.

a1  a2

 a3 b1   a4=b2  a5=b3

 a6=b4

the pseudo lattice graph of type I by merging vertices a4 with b2, a5 with b3 and vertex b4 with a6. This is again a tree.

Example 2.6: Let L1 be a lattice.

a1 

a2  a3 a4  

 a5

 a6

60 Pseudo Lattice Graphs and their Applications to Fuzzy…

and L2 a lattice  b1

 b2 b b 3   4

 b5

 b6

L1 is a modular lattice and L2 is a distributive lattice.

We can merge vertex b4 with vertex a2 and get the following pseudo lattice graph of type I.

 b1 a1   b2 b3  a2  b4  a3  a4

  b5 a5

 b6  a6

This is a graph and not a lattice.

We can merge vertex a4 with b1 and get the following pseudo lattice graph of type I.

a1 

a2  a3 a4=b1    a5  b2  b3   b4 a6  b5

 b6 Pseudo Lattice Graphs of Type I 61

This is a graph and not a lattice.

 b1

 b2 b4 b3  a1  

 b5  a2  a4

 b6  a5  a6

Let the vertex b5 be merged with a2.

The pseudo lattice graph of type I is only a graph.

We can merge vertex b6 with vertex a1 and get the following pseudo vertex graph of type I.

 b1

 b2 b3   b4

 b5

 a1 = b6

a a 2   a3  4

 a5

 a6

This is a lattice.

We can merge vertex a6 with b1 which is a pseudo lattice graph of type I which is the following lattice.

62 Pseudo Lattice Graphs and their Applications to Fuzzy…

a1 

a2   a3  a4

 a5

 a6 = b1

 b2

b b 3   4

 b5

 b6

Thus we can get pseudo lattice graph of type I using the lattices L1 and L2.

Example 2.7: Let L1 =

a1 

a2   a3

a  4

a5   a6

 a7

a 8  a9

 a10

Pseudo Lattice Graphs of Type I 63

and L2 =  b1

 b2

 b3

 b4

b5   b6

 b7

be any two lattices.

We can merge vertex b1 with a2 and get the following pseudo lattice graph of type I.

a1 

 b1=a2  a3  b2  a4  b3

 b4 a5   a6

b b 5   6  a7

 a8   a9 b 7  a10

This is only a graph and is not a lattice.

Let us merge vertices a3 with b2 and b3 with a5 we get the following pseudo lattice graph of type I.

64 Pseudo Lattice Graphs and their Applications to Fuzzy…

a1   b1

a2   a3=b2

 b3  a4  b4

a5   a6 b5   b6

a  7  b7 a8   a9

 a10

We see this not a lattice only a graph.

We can also merge only three vertices and get the pseudo lattice graph of type I which is as follows.

a1 

a2   a3 = b1

 a4

a5   a6= b2

 a7

a8   a9= b3

 a10

 b4

b5   b6

 b 7

This is not a lattice only a graph.

Pseudo Lattice Graphs of Type I 65

Thus by this new method we get several types of graphs which are new and enjoy properties like being a lattice and so on.

We can also merge only four of the vertices and get the following pseudo lattice graphs of type I.

a1= b1 

a2   a3

 a4= b2

a 5   a6

a = b  7 3

a8   a9

 a10=b4

b 5  a6



b7 which is a graph.

Example 2.8: Let us consider two Boolean algebras.

a1 

B1 = a2   a3

 a4

66 Pseudo Lattice Graphs and their Applications to Fuzzy…

and b1 

b3 B2 = b 2   b4

b    b 7 b6 5

 b 8

We find the pseudo lattice graphs of type I by merging the vertices or edges or both. It is important to keep in record is that it will not give a new Boolean algebra.

The pseudo lattice graph of type I may be a lattice or a graph a Boolean algebra when the Boolean algebra B1 is merged with B2 in such a manner that all the four vertices are merged with four vertices and four edges are also merged with four edges. Then we get the Boolean algebra B2 only.

By merging vertex a4 of with vertex b1 of b2 we get the pseudo lattice graph of type I which is as follows:

a1 

a2  a3

 a4 = b1

b3 b 2    b4

   b7 b b5 6  b8

This is a lattice which is distributive and not a Boolean algebra. Pseudo Lattice Graphs of Type I 67

By merging a1 with b8 we get the following pseudo lattice graph of type I. This is also not a Boolean algebra only a lattice.

b1 

b3 b2   b 4

b    b 7 b6 5

 b8= a1

a2  a 3  a4

Let us merge vertices a3 with b2 and get the following pseudo lattice graph of type I.

b1 a1   b3 a2  a3  b2   b4

  b7   b5 a b6 4  b8

Clearly this is only a graph and not a lattice.

The merging of vertices can be b5 and a2 or b4 a2 or b7 and a2 so on.

Now we can also merge the edges a1a2 with b5b8 and vertices a1 with b5 and a2 with b8 which is as follows: 68 Pseudo Lattice Graphs and their Applications to Fuzzy…

b1  b3 b2    b4

   b7 b6 b5= a1

 a4  b = a 8 2  a4

The pseudo lattice graph of type I is a lattice.

Suppose vertex b4 is merged with a1 and b5 with a2 we get the following pseudo lattice graph of type I.

b1 

b3 b4= a1 b2     a3

b    b = a  a4 7 b6 5 2



b8

Now we can merge edge b1b3 with a1a2 in the following way and get a pseudo lattice graph of type I.

b1  a3  b a5 b 3 2    b4

   b7 b6 b5

 b8 Pseudo Lattice Graphs of Type I 69

Finally we can merge four vertices and four edges in the following way to get the pseudo lattice graphs of type I.

a1 = b1 

b3 a2=b2    b4 = a3

   b7 b6= a4 b5

 b8

The resultant is a Boolean algebra B2.

Thus by merging like this in six ways we get only a Boolean algebra of order 8 that is B2 itself.

Now we can merge only two of the vertices b6 with a2 and b5 with a3 and

b1 

b3 a1 b2    b4

   b7 b6= a2 b5 = a3

  b8 a4

The pseudo lattice graph of type I is not a lattice it is only a graph.

We can merge vertex b6 with a1 and get the following pseudo lattice graph of type I. 70 Pseudo Lattice Graphs and their Applications to Fuzzy…

b1  b b 3 2    b4

  b7 b b5 6

a2    a3 b8  a4

Likewise we can merge vertex b3 with vertex a4 which gives the following pseudo lattice graph of type I.

a1 

 b1 a2  a3 b3 b 2    b4 a4    b7 b6 b5

 b8

Example 2.9: Let us consider two lattices

 a1

 a2

a a L1 = 4   3

 a5

a  6

Pseudo Lattice Graphs of Type I 71

b 1

b3  L2 =  b2

b4   b5

 b6

We can adjoin the vertex a6 with b1 which is as follows:

 a1

 a2

a4   a3

 a5

a6=b1

b3   b2

b4   b5

 b6

The pseudo lattice graph is a lattice which is both non distributive and non modular.

72 Pseudo Lattice Graphs and their Applications to Fuzzy…

b1 

b3   b 2

b4   b5

 b6=a1

 a2

a4   a3

 a5

a  6

The pseudo lattice graph is a distributive lattice.

However by merging vertex a1 with vertex b5 we get the following pseudo lattice graph of type I.

b 1

b3   b2

b4   b5 = a1

 b a2 6 

a4   a3

 a5

 a6

Pseudo Lattice Graphs of Type I 73

This is only a graph and not a lattice. We can merge edge a1a2 with b2b5 which is as follows:

b1 

b3   b2=a1

b4   b = a 5 2 b6   a3  a4

 a5

a  6

This is a pseudo lattice graph of type I which is not a lattice.

We can merge the edge a1a2 with b2b5 and edge a2a4 with b5b6 which is as follows:

b 1

b3   b2=a1

b4   b5 = a2

  a3 b =a 6 4  a5

 a6

Clearly the pseudo lattice graph of type I is a lattice. We can merge edges a5a6 with b2b5 and edge a6a5 with edge b1b2 and get the following pseudo lattice graph of type I.

74 Pseudo Lattice Graphs and their Applications to Fuzzy…

 a1

 a2 b1=a4   a3

 b3  b2=a5

b4   b5 = a6

 b6

This is also a lattice. We can also merge b1b3 with a2a4 and b1b2 with edge a2a3 and get the following pseudo lattice graph of type I.

 a1

 a =b 2 1

a4=b3   a3=b2 a5  b4  b5 a6  b6

The resulting pseudo lattice graph of type I is only a graph and not a lattice.

In the same pseudo lattice graph of type I we can merge also vertex a6 with vertex b6 and get the following pseudo lattice graph of type I.

Pseudo Lattice Graphs of Type I 75

 a1

 b = a 1 2

a4=b3   b2 = a3

a5  b4  b5

 b6=a6

This is a lattice or order 8.

We get the following pseudo lattice graph of type I.

 a =b 1 1

 a2 b3  b2   a4 a3  a5  b4  b5

 b6=a6

Thus this is only a graph and not a lattice.

76 Pseudo Lattice Graphs and their Applications to Fuzzy…

 a1

 a2=b1

b3  b2 a3  a4 

b4  b5

 b6=a5

 a6

This pseudo lattice graph of type I is a lattice.

a1=b1 

b2  b3

b5  b4

 b6=a2

a4   a3

 a5  a6

This is a lattice as well as a graph.

Pseudo Lattice Graphs of Type I 77

Example 2.10: Let L1 =  a1

a2 

 a4 a3 

 a5

 a6

a8   a7

 a9

b1 and L2 = 

b2 

 b4

b  3



b5 be two lattices. We can merge a4a5 with b2b3 and get the following pseudo lattice graph of type I.

 a1

b1  a2 

 b2= a4

a3   b4

 a =b 5 3

 a6  b5

a8   a7

 a9 78 Pseudo Lattice Graphs and their Applications to Fuzzy…

This is only a graph and not a lattice.

We can also merge a2a3 a4a5a6 part with b1b4 b2b3b5 and get a pseudo lattice graph which is as follows:

It is clearly a lattice and that lattice is L1.

 a1

 a2=b1

 a4=b2 a3=b4   a5=b3

 a =b 6 5

a8   a 7  a9 We can also merge the vertex a7 with v1 and edge a7a9 with b1b4 and b4 with a9 and obtain a pseudo lattice graph of type I which is as follows:

 a1

 a2

 a4  a3

 a5

 a6

a =b a8   7 1

  b2 b4=a9

 b3  b5 Pseudo Lattice Graphs of Type I 79

This pseudo lattice graph of type I is a lattice.

Example 2.11: Let L1 and L2 be the two lattices given by

 b1 a1  b2  and  b3 a2   a 3  b4  a4

a  5 b5  b7  a6 b6  a7  b8

We can merge the edge a6a7 with edge b1b2 and obtain the following pseudo lattice graph of type I.

a1 

a2   a3

 a4  a5

 a6=b1

 a7=b2

 b3  b4

b5

 b 7  b6

 b8 80 Pseudo Lattice Graphs and their Applications to Fuzzy…

Clearly this is a lattice which is not distributive and not modular.

Now we merge b4 with a1 and b8 with a4 and obtain the pseudo lattice graph of type I.

 b1  b2

 b3

 b4=a1

b5 a2  a3   b7 b6

 b8=a4 a  5  a6

 a7

Clearly the resultant is a lattice and not a modular or distributive lattice.

We can merge the vertex a2 with b7 and obtain the pseudo lattice graph of type I which is as follows: Pseudo Lattice Graphs of Type I 81

 b1

 b2

 b3

b  4 a1  b5

 b7=a2  a3

b6  a4

  a5 b8  a6

 a7

Clearly the pseudo lattice graph is a lattice.

We can merge vertex b2 with a6 and get the following pseudo lattice graph.

a2  a4  b1 a5 a1     b2=a6 a7  

a3  b3

 b4

b5

 b7

b  6



b8

82 Pseudo Lattice Graphs and their Applications to Fuzzy…

The resultant is only a graph and not a lattice.

Now having seen several examples of pseudo lattice graphs we define substructure in them.

DEFINITION 2.1: Let L1 and L2 be two lattices. S be the pseudo lattice graph of type I obtained by merging a vertex or more vertices or merging edges and vertices. Let P be the subgraph of S; subgraph defined in the usual way, then

(i) P can be lattice or (ii) P can be a graph.

P is defined as the pseudo lattice subgraph of S of type I.

We will illustrate this situation by an example or two.

Example 2.12: Let L1 and L2 be the following lattices.

a1 b1   a a 3 1    a4 b2 and  b3  a7   a5 b  a6 4

 

a8 b5

Let S be the pseudo lattice graph of type I obtained by merging a4 with b2 and b4 with a5.

a1 b1  

a3 a1    a4=b2  b3    a7 a6 a5=b4

  a8 b5 Pseudo Lattice Graphs of Type I 83

Clearly S is not a lattice S has several subgraphs that are subgraphs and not lattices and a few lattices.

We will illustrate this in the following.

a1  a1 b1  

a1   a4=b2

  a7 a6  b4=a5    a8 b5 a8

P1 is a subgraph of S which is not a lattice.

P2 is also a subgraph of S which is not a lattice P2 is a tree.

Consider P3 the pseudo lattice graph of type I which is as follows. a1 b1   a =b 4 2  a2   b3  a7  a5=b4   a8 b5 is only a subgraph which is a connected subgraph of S.

Let P4  a1  b 1

a2  a3  a4=b2  b3  a6

84 Pseudo Lattice Graphs and their Applications to Fuzzy…

be the pseudo lattice subgraph of type I.

P4 is a graph and it is not connected.

 a2 Let P5 a2   a7

a7 

be the pseudo lattice graph of type I. Clearly P5 is a lattice.

Thus with S got by merging an edge of the two lattices L1 and L2 we got 5 subgraphs of S.

Now we obtain a pseudo lattice graph of type I by merging the edges a2a7 with b2b4 edge a1a2 with edge b1b2 and edge a1a4 with a1b4 which is denoted by S1 is as follows:

a1=b1  a a 3 1    a4=b3

   b4= a7 a a5 6

  b5

a8

Clearly S1 is not a lattice only a graph 9 vertices and 14 edges.

However this is only a graph.

Example 2.13: Let L1 and L2 be two lattices given in the following. Pseudo Lattice Graphs of Type I 85

 b1

a1  b2  a4 L1= a2   a3  a4 L2 = b3 

  b5 a5

 b6

Let S1 be the pseudo lattice graph obtained by merging the edges a3a5 with b1b2

a1 S1 =  a3=b1 a2    a4

 a5=b2

b3  a4

 b5

 b6

Clearly the pseudo lattice graph of type I. S1 is a lattice.

S1 is modular and non distributive.

a1  P1 = a2   a4

 a5=b2

b3   b4

 b5

86 Pseudo Lattice Graphs and their Applications to Fuzzy…

is a subgraph which is distributive lattice.

a1  P2 = a2   a4  a3

 a5

b3   b4

is a subgraph which is a tree. P2 is not a lattice.

Consider P3

a1 

a2   a4  a3

 a5

b 3   b4

 b5

 b 6

a pseudo lattice subgraph of S1. P3 is not a lattice only a graph which is a tree.

Thus S1 has several subgraphs which are lattices or graphs which are trees or otherwise.

Let us consider S2 the pseudo lattice graph given by the following.

Pseudo Lattice Graphs of Type I 87

 b1

 b2 S2 = b   a4 3  a1 b5 

a2=b6   a3  a4

 a5

S2 is got by merging a2 with b6. We see S2 has subgraphs some of which are sublattices and some of them are subgraphs.

P1 =  b1

 b2

 a4

 b5

b6 

 a5

 b1

 b2 P2 =

b3  a4

 b5

a1 P3 = b5 

b6=a2  a3  a4

 a5 88 Pseudo Lattice Graphs and their Applications to Fuzzy…

pseudo lattice subgraphs some of which are sublattices and some are subgraphs.

b5

a b6=a2  a3  4  a5

is a pseudo lattice subgraph which is a subgraph and not a lattice.

Infact P3 can be realized as a semi lattice.

However S3 is connected.

P4 =

 a1 b3  b4 a2=b6  a3=b1   b a5 5

be a pseudo lattice subgraph of S2. P4 is only a subgraph which is not connected.

Let M be a pseudo lattice graph got by merge edge a1a2 with edge b2b3, edge a1a4 with b2b4 and edge a2a5 with b3b5 and edge a4a5 with b4b5 and get the following graph;

Pseudo Lattice Graphs of Type I 89

b1

 b2=a1

b = a b6=a2  a3  4 4  b5=a5

 b6

This is a lattice we have the following pseudo lattice subgraph of M.

b1

 b2=a1

b6=a2  a3  b4= a4

 b5=a5

 b6

which is a tree and not a lattice.

Consider the subgraph

b3=a2  a3  b4= a4

 b5=a5

b  6 which is also a subgraph is a tree.

90 Pseudo Lattice Graphs and their Applications to Fuzzy…

Now having seen merging vertices or edges of two lattice we now proceed onto merge three vertices of three lattices as a single point or merging vertices taken two by two of lattices, the same is true for edges.

This merging will result in a graph or a lattice known as multi pseudo lattice graph.

This will be illustrated by the following examples.

Example 2.14: Let

L1 =  a1

a2   a3

 a4

 a5

a7   a6

 a8

c1  b1 L2 = and L3 = c2   b2  c4 b3  b4 b5 c3 

 b 6  c5

be three lattices by merging b1 with c1 and a8 vertices we have the following pseudo lattice graph of type I.

Pseudo Lattice Graphs of Type I 91

 a1

a2   a3

 a4

a  5

a7   a6

c1 b1 a8

c2   b2

 b5=c4 b3  b4   c3  b 6  c is not a lattice only a graph. 5

Now we merge the three edges a6a8, b2b2 and c4c5 and get the following pseudo lattice graph of type I.

a1 

a2   a3

 a4

a b 5 c1 1  a7  c2  a6=b2=c4 c3   b4 b5

b3=a8=c5  b6

Clearly the resultant is only a multi pseudo lattice graph.

We can by this way build several such graphs.

92 Pseudo Lattice Graphs and their Applications to Fuzzy…

Example 2.15: Let us consider the three chain lattices.

L1=  a1 L2 =  b1 c1  a2  b2  c2

 a3  b3  c3

a b c  4  4  4  a5  c5

 a6  c6

 c 7

By merging the vertices a4, b3 and c6 we get the following multi pseudo lattice graph of type I.

a1 c1  c2 a  2  c3 b 1  a3  c4  c5 b 2  a4=b3=c6  a5  b4  c  a 7 6

is not a lattice only a graph in fact a tree.

We can also merge in the following way.

c 1 c  2 a1  c3  a2=b2=c4

 a3=b3=c5

 c6 a4  b4  a5  c7  a6 Pseudo Lattice Graphs of Type I 93

We can merge vertices a1, b1 and c1 together

a1 = b1= c1   c2  a2  b2  c3

 a3  b3  c4

 a4 b4 c   5 a  5  c6

 a6  c7

The pseudo lattice graph of type I is not a lattice only a graph which is a tree.

Example 2.16: Let L1, L2 and L3 be three lattices given in the following. a1 b1   L1 = L2 =

b2   b3 a2   a3  b4 a6   a4 b5   b6

   c1 a5 b7

c   c3 2

 c4

 c5

and L3 = c7 

 c6

 c8



c9 94 Pseudo Lattice Graphs and their Applications to Fuzzy…

We merge edges a3a4, c4c5, a1a3 with c2c4 and a4a5 with c4c7 and vertices b2 with a3 and b5 with a4 and get the following multi pseudo lattice graph of type I.

c1 

a1=c2   c3=b1

a  b3 2   a3=c=b2

 b4 a6  

a4=c5=b5 a5 = c7   b6

 c6=b7 c  8  c9

We can merge vertices a3 with b2 and a4 with b5 and b3 with c2 and b6 with c7 and get the pseudo lattice graph of type I.

a1  b1  c1  a3=b2 a2    c2  c3 b6   c4  b a6  a4 6   c5 c7   c6  b7  c8 a5

 c9

Pseudo Lattice Graphs of Type I 95

The resultant graph is only a graph and not a lattice. However it is a connected graph.

 a1 Example 2.17: Let L1 = a2   a3

 a4

 a5

c1 b1  L2 = L3 =  b2 c3 c2     b3 c4  b4   c7  c c5  b5 6

 c8 be three lattices let us merge b1b2 with c2c7 and c4c5 with a4a5 c1c4 with a2a4 and the resulting pseudo lattice graph of type I, S1 is as follows.

a1  c =a 1 2   a3

c3 a1 c2   c =a 4 4

c  b2   c =a 7 c6 5 5 b3   b4  c8 b5  96 Pseudo Lattice Graphs and their Applications to Fuzzy…

Let us merge b1b2 with c2c7 and c4c5 with a4a5, c1 c4 with a2a4 and the resulting pseudo lattice graph of type I, S1 is as follows.

Clearly S1 is not a lattice only a graph.

We can merge the 3 vertices b1, c5 and a2 and get the following multi pseudo lattice graph of type I.

c1 

c3 c2   c4  a1

   c5=b1=a2  a3 c7 c6  b3  a4   b3 c8  b4  a5

 b5

This is only a graph and not a lattice.

 a1 Example 2.18: Let L1 = a2  a5 a3

 a4

a6  b1

 b2  c1 c4 b3   b4 L2 = L3=   c3 c2

 b5 c6  c5

c7  b6 Pseudo Lattice Graphs of Type I 97

Merge vertices c6 with b6 and c2 with a6, we get the following multi pseudo lattice graph of type I which is as follows.

b  1  a1  b2 a2  a5 a3

  b4 b3   c1 a4 c4  b5 c   ca36 2

c6 b6  c 5 c7

The resulting diagram is only a graph and not a lattice.

Now we can for all these pseudo lattice graphs of type I find subgraphs and the study the property of connected ness and so on.

 a1  b1

 a2  a3  b2

 c1

 c4  c3

 c7 is a subgraph which is not connected.

 a1

a2  a5 a3

 a4 a subgraph which is a lattice which is connected. 98 Pseudo Lattice Graphs and their Applications to Fuzzy…

Now having seen examples of pseudo lattice graphs of type I and their substructures we proceed onto define pseudo lattice graphs of type II using a lattice and a graph.

Chapter Three

PSEUDO LATTICE GRAPHS OF TYPE II

In this chapter the new notion of merging one or more vertices of a lattice with that of a graph or one or more edges of a lattice with a graph is carried out. This study is new and innovative. The resultant graph (or lattice) is defined as the pseudo lattice graph of type II.

In the earlier chapter merging of a vertex or more vertices or one edge or more edges of lattices was carried out. Those resulting graphs or lattices were defined as pseudo lattice graph of type I. Several interesting features about these pseudo lattice graphs of type I was systematically defined and developed.

Before we make the definition of pseudo lattice graph of type II we will first illustrate the situation by an example or two. 100 Pseudo Lattice Graphs and their Applications to Fuzzy…

a1 Example 3.1: Let L be the chain lattice of order two;  v1   a2 Let G be the graph   v v2 3  v4

We give some of the possible merging of the vertices.

a1  a1=v1  a2=v1 a2    v2 v3   v v2 3 

 v4 v4

v1 

 v  v3 2  v4 = a1

 a2

v1   v  a1  v3 2

v4 = a2 Pseudo Lattice Graphs of Type II 101

v1 

 v =a  a2  v3 2 1

v4

v1 

v2    a1 v3=a2

v 3

v1 

v2   v3=a1

  a2 v4

v1 

a1=v2   v3

 a2  v4

v1=a1 

v2   v 3

v4=a2

102 Pseudo Lattice Graphs and their Applications to Fuzzy…

v1=a1 

v2   v3

v =a 4 2

The last two pseudo graphs are got by merging two vertices.

Finally we can merge an edge and two vertices.

So that we get

v1 

v2   v3=a1

 v4=a2

Such merging will be called as special trivial merging for the resultant gives the graph G (it may give the lattice L).

Finally we can merge two vertices so that the graph has the following form

v1=a1 v1=a1  

v2=a2   v3 v2   v3=a2

 v 4 v4

Pseudo Lattice Graphs of Type II 103

Thus we can get several pseudo lattice graphs of type II.

v1 v1  

v2   v3=a1 a1=v2   v3

  v4=a2 v4=a2

Example 3.2: Let L be a lattice given by

 a1

a3   a2 a4   a 5 and G be the graph given in the following.

v1 v 2   v 3 v4   v5

 v6

We find pseudo lattice graphs of type II.

Merging of one vertex of L with one vertex of a graph G. 104 Pseudo Lattice Graphs and their Applications to Fuzzy…

 a1

a3 

 a 2 a4   a5=v1 v2   v3

v4   v5  v6

 a1 a3 

va12

a4  v2  a5  v3

v4   v5  v6

 a1

a3 

 a2

a4 v1  v2   v3 a5

v4   v5  v6 Pseudo Lattice Graphs of Type II 105

 v1=a1 v2 a3    v3

v4   v5  a2 a4   v6  a5

 a1 a3=v1 

 a2 v2   a4   v5  a5 v4 

 v 6

Example 3.3: Let v1  v2 v5  

G =  v3 be a graph  v4

and L =  a1

a3 

 a 2 a4   a5 106 Pseudo Lattice Graphs and their Applications to Fuzzy…

be a lattice. The pseudo lattice graph of type II got by merging v1 with a5 is follows:

 a1 a3 

 a2

a4  v1=a5  v2 v5  

 v3  v4

The resultant is a graph we merge the vertices v2 with a5 and v3 with a4 and obtain the pseudo lattice graph of type II. v1  v5  v2=a5   a1

 v4  v3=a4

 a2

 a3 The resultant is only a graph.

We see both the graphs are distinct. We can merge the 5 vertices v1, v2, v3, v4 and v5 with a1, a2, a3, a4, a5 which is as follows.

Pseudo Lattice Graphs of Type II 107

v1=a1  v2=a2 v5=a5  

 v3=a3 v4=a4 

This is a graph.

Thus the pseudo lattice graph of type II got by using L with G is a complete graph.

Example 3.4: Let G =

v1 

v2  v3

 v4

  v6 v5

a 1

a and L = a2  a3  4  a5

be the graph and lattice respectively.

We can merge vertices a5 with v1 and obtain the following pseudo lattice graph of type II. 108 Pseudo Lattice Graphs and their Applications to Fuzzy…

a1 

a2  a3  a4

 v1=a5 v2  v3

 v4   v6 v5

This is only a graph.

Now we merge v1 with a1, v2 with a2, v3 with a4, v4 with a3 and obtain the following pseudo lattice graph of type II.

a1=v1 

v =a v2=a2  4 3 a4=v3

 a5  

v6 v5

The resultant is again a graph. We can also get the pseudo lattice graph of type II by merging edges v1v2 with a1a2 and v1v3 with a1a4 and v1v4 with a1a3 and is as follows.

v1 = a2 

a3 = v4 a2=v2   v3=a4



a5   v6 v5 Pseudo Lattice Graphs of Type II 109

The resultant is only a graph and not a lattice.

Example 3.5: Let

G = v1 v2

v4  v 3

a1  and L = a2  a3  a4

be a graph and lattice respectively.

By merging vertex a1 with vertex v3 we get

v 1 v2

v4  v3=a1

a2  a3

 a 4

The pseudo lattice graph of type II is only a graph.

We can merge the edge v2v4 with a1a2 and obtain the following pseudo lattice graph of type II.

110 Pseudo Lattice Graphs and their Applications to Fuzzy…

v1  v2=a1

v4=a2  v3 a3

 a 4

This is only a graph. We can merge the four edge and four vertices and get the following pseudo lattice graph of type II.

v1=a1

v =a v3=a3   2 3

 v4=a4

which is nothing but the graph G.

By merging vertices v1 with a1 and v3 with a4 we get the following pseudo lattice graph of type II.

v1  v2=a1

a a2  3

v4  v =a 3 4 Pseudo Lattice Graphs of Type II 111

The resultant graph is only a graph.

Example 3.6: Let L =  a1

 a2

 a3

 a 4  a5 and  v1

v6   v2 G =

v 5  v3  v4 be a lattice and a graph respectively.

We get the following pseudo lattice graphs of type II.

 a1

 a2

 a3

 a4

 a5=v1

v6   v2

v5   v 3  v4

112 Pseudo Lattice Graphs and their Applications to Fuzzy…

 v1  a1

v6   v2=a2

P2 =

v5   v3=a3

  a4 v4

 a5

and

 a 1  a2=v1

v6   v2

P3 =

v5=a3  v 3  a4  v4  a5

All of them are only graphs.

Example 3.7: Let a 1

a2  a4

a3   a5 Pseudo Lattice Graphs of Type II 113

be the pentagon lattice and

v1 

 v6 v10 v v    7  2 v5 v8 v9   v3  v 4 be a graph. We can get several pseudo lattice graphs of type II which are as follows:

v1 

 v6 v10 v v    7  2 v5 v8 v9   v  3 v4 is a graph.

v1=a1 

 v6 v10 a2=v5 v7  v2=a4   

v8 v9   v =a  3 5 a3=v4 114 Pseudo Lattice Graphs and their Applications to Fuzzy…

We get the pseudo lattice graph of type II to be a graph which is the Peterson graph.

v6 

 v1 v5   v2  v7 v10 v3=a1 a2=v4  

  v8=a4 a3=v9

 a5

is a pseudo lattice graph of type II obtained by merging the vertices a1 with v3, a4 with v8, a8 with v9 and a2 with v4 and merging the edges v1 v9 with a2 a3 and v3 v8 with a1 a4.

The resultant is only a graph.

v1 

 v6 v10 v7  v2    v 5 v8 v9   v3=a1 v4=a5

  a2 a4

 a3

Pseudo Lattice Graphs of Type II 115

The merging of the vertices v3 with a1 and v4 with a5 results in a pseudo lattice graph of type II which is a graph.

Example 3.8: Let

a1 

a2   a3

L =  a4 a 5   a6

 a7

and v1 

   v v2 v3 4 G = v7  v   v5 6 v8   v9

be a lattice and a graph respectively.

We can merge vertex v8 with a2 v6 with a1 and v9 with a3 and obtain the following pseudo lattice graph of type II.

116 Pseudo Lattice Graphs and their Applications to Fuzzy…

v1 

 v v  v4 2 3 v6=a1 v7    v5

v8=a2   v9=a3

 a  4 a 5   a6

 a7 Clearly this is not a lattice only a graph.

We can also merge in the above the edges a1a2 with v6v8, a1a3 with v6v9 and get the following pseudo lattice graph of type II.

v1 

 v v  v4 2 3 v6=a1 v7    v5

v8=a2   v9=a3

 a4

a5   a 6  a7

We can merge only the vertex a7 with v1 and get the following pseudo lattice graph of type II. Pseudo Lattice Graphs of Type II 117

a1 

a  2  a3

 a4

a 5   a6

a7=v1

  v v2 v3 4

v   v5 7 v6 

v8   v9

 a Example 3.9: Let L = 1  a2

 a3

 a4

 a 5  a6

 a7 and G = v1  v12   v2 v13   v11  v10  v3 v9 v3

v8  v5

 v7  v6 118 Pseudo Lattice Graphs and their Applications to Fuzzy…

be a lattice and graph respectively where ever adjoin a vertices of G with L we will get only the pseudo lattice graph of type II to be only a tree.

v1=a1  v12   a2  v2 v13   v11  v10   a3 v3  v9   a4 v3 v8  v5  a5  v7  v6  a6

 a7

However if we merge two vertices of the graph with the lattice we may not in general get a tree.

This is illustrated by the following.

v1  v12   v2 v13   v11  v10 v3 v9 v3

a1= v8   v5

v7=a2   v6 a3 a4 a5 a6 a7 Pseudo Lattice Graphs of Type II 119

Clearly this pseudo lattice graph of type II is not a tree only a graph.

Let us merge the vertices v10 with a1, v9 with a2, v8 with a3 and v7 with a4. The following pseudo lattice graph of type II is not a tree.

v1  v12   v2 v13   v11 

a1=v10 v3  v9  v3 a3= v8   v5  a4=v7  v6  a5 a6  a7

Clearly this is only graph and not a tree.

So merging a tree with a chain lattice may not in general give a pseudo lattice graph of type II which is a tree.

a1 Example 3.10: Let L = 

  a  a4 a2 3

a5  a6

120 Pseudo Lattice Graphs and their Applications to Fuzzy…

and v2  v1   G = v3

v4 v 5   v6

be the lattice and graph respectively. We can merge v3 vertex with a4 and obtain the pseudo lattice graph of type II. v2   a1=v3  v1

  a  a3 a4 2 v  4 a5 v 5   v6  a 6

This is only a graph.

a1 Example 3.11: Let L = 

a3 a2    a4

a    7 a6 a5

 a8 Pseudo Lattice Graphs of Type II 121

and  v1 v2 v3 G =    v4 be a lattice and graph respectively.

By merging vertices v3 with a3 and edge v2v3 with a1a3 v1v3 with a3a5 and v3v4 with a3a7 we get the following pseudo lattice graph of type II which is as follows:

a1=v2 

a3=v3 a2    a 4

a =v    a =v 7 4 a6 5 1  a8

This is a lattice which is a Boolean algebra.

a1  a a 3 2    a4

   a7 a6 a5

a8

 v3

  v4 v1 122 Pseudo Lattice Graphs and their Applications to Fuzzy…

The pseudo lattice graph of type II is only a graph and not a lattice.

By merging the edge a4a5 with v2v3 we get the following pseudo lattice graph of type II.

a1 

a a 3 2    a4=v2

  a5=v3 a7  a 6    v4 v1 a8

This is only a lattice.

Suppose a5a8 with edge v3v4 in addition to merging the edge a4a5 with v2v3 we get the following pseudo lattice graph of type II.

a1 

a3 a2    a4=v2

   a5=v3 a7 a6   v1 a8=v4

This is a graph and not a lattice.

Pseudo Lattice Graphs of Type II 123

Example 3.12: Let L be the lattice

a1 

a 3   a2

a4   a5

 a6

G =    v1 v2 v3 and G the graph.

What ever be the merging a vertex or two vertices we will get only the pseudo lattice graph of type II to be a graph.

We merge vertex a1v1 and a6 and v3 and obtain the following pseudo lattice graph of type II.

a1=v1 

a3   a 2

a   v2  a 4 5  a6=v3

124 Pseudo Lattice Graphs and their Applications to Fuzzy…

This is only a graph and not a lattice.

Example 3.13: Let

G =   v2 v1

be a graph and L is a lattice given in the following.

a1

a2

   a3 a5 a4

a 6   a8 a7

a9



a10

Any pseudo lattice graph of type II by merging any of the vertices or edges is only a graph and never a lattice as the graph has self loops.

In view of this we have the following theorem.

THEOREM 3.1: Let L be any lattice and G a graph with a loop. The pseudo lattice graph of type II using this L and G is never a lattice.

Proof: Follows from the fact G is a graph with a loop we see so the pseudo lattice graph of type II can never be a lattice. Pseudo Lattice Graphs of Type II 125

Example 3.14: Let L be a lattice whose Hasse diagram is as follows: a 1

    a  a6  a7 a2 a3 a4 5  a8 and G be the following graph.

 v1

  v2 v3

Let the pseudo lattice graph of type II be obtained by merging vertices a2 with v2.

We get the following graph.

 v1 a 1

     a  a6  a7 a3 a4 5 v3 a2=v2  a8

We can merge the edges a8a7 with v1v3 and get the following pseudo graph of type II. 126 Pseudo Lattice Graphs and their Applications to Fuzzy…

a1  a7=v1      a6  a2 a3 a4 a5

 a =v 8 3

 v2

Clearly this is only a graph.

Example 3.15: Let L be a chain lattice

 a 1 C4 =  a2

 a3

 a4

and G be a tree

v1 

 v2  v3

 v v7  4     v v8 v9 6 v5

Lattice graph of type II is a tree in some cases only not in all cases.

Pseudo Lattice Graphs of Type II 127

Suppose we merge vertices v1 with a1 and a4 with v6 we get the following pseudo lattice graph of type II.

v1=a1 

 v  a2  v3 2

v7   a3  v4

    v8 v9 a4=v6 v5

The resultant graph is only a graph and not a tree.

Let us merge vertex v3 with a2 and v9 with a4 and get the pseudo lattice graph of type II which is as follows:

a1 v1  

 v2  v3=a2 a3   v v7  4     v v8 a4=v9 6 v5

We get a graph which is not a tree.

Let the pseudo lattice graph of type II of L and G got by merging the vertices v1 with a1 v2 with a2, v7 with a3 and v8 with a4 be obtained which is given in the following.

128 Pseudo Lattice Graphs and their Applications to Fuzzy…

v1=a1 

 v2=a2  v3

v3=a3   v4     v8=a4 v9 v6 v5

The resultant is a graph identical with G hence a tree.

We will merge vertex v4 with a1 the resultant pseudo lattice graph of type II is as follows:

v1 

 v2  v3

 v4 = a1 v7      v v8 v9 6 v5  a2

 a5  a4

We see the resultant is a tree different from G.

Now we will see the substructures of the pseudo lattice graphs of type II.

This is illustrated by the following examples.

Pseudo Lattice Graphs of Type II 129

Example 3.16: Let

 a1

 a 2 L =  a3

 a4

 a5  a6 a chain lattice and G be the graph

v1 

v2   v6

v4  v3 v9  v7

v  v 5 v10 8   v v 12 11

be a tree.

The pseudo lattice graph of type I in general is not a lattice or a tree only a graph.

We find subgraphs of the pseudo lattice graphs of type II.

130 Pseudo Lattice Graphs and their Applications to Fuzzy…

v1 

v2  a  2 P1 =  v6   a3 v4 v3  v 9 v7  a4 v5  v8 a5 v10    a6  v 12 v11

This is a tree and every subgraph of this P1 is also tree.

We have the following pseudo lattice graph of type II.

 a1

 a2 P2 =

 a3=v1

v2  a4  v a  6  5  v4 v3  v 9 v7 a6 

v5  v v8 10   v v 12 11 All subgraphs of P2 are only trees in this case also.

Let us merge vertices v8 with a1 and v12 with a2 and get the pseudo lattice graph of type II which is as follows:

Pseudo Lattice Graphs of Type II 131

v1 

v2   v6

v4  v3 v9  v7=a1

v5   v8 v10   v11=a2 v 12  a3

 a4

 a5

 a6

We see this is not a lattice a graph which is not a tree.

This has subgraphs which are trees for instance

v1 

v2   v6 v  4 v3 v8   v7

v5 

Subgraphs which are chain lattices viz.

132 Pseudo Lattice Graphs and their Applications to Fuzzy…

 v1

 v6=a1

 v12=a2

 a3

 a4

 a5

 a6

Now consider the subgraph

 v6

v7 

 v8  v10  v12=a2

This subgraphs is non modular and a non distributive lattice.

Pseudo Lattice Graphs of Type II 133

The subgraph

 v6

v7   v8

v10   v12=a2

 a3

 a4

is again a non modular non distributive lattice. All these subgraphs are connected.

We have subgraphs which are disconnected also. These are illustrated in the following.

v 1  v6 

v2 a1=v8   a4 

v4   v3 a  5 v9 

a6 

The above subgraph is a not connected subgraph.

Consider the following subgraph. v2 

v2  a  v5   v6 2

v7  v8=a1

v10 134 Pseudo Lattice Graphs and their Applications to Fuzzy…

This is also a subgraph which is not connected.

Example 3.17: Consider the following lattice

a1 

a2   a3

 a4

a5   a6

 a7

 a8

and the graph

v1  v2 v5  

 v3  v4

The pseudo lattice graph of type II of the lattice and graph by adjoining two vertices is as follows:

Pseudo Lattice Graphs of Type II 135

a1 

a2   a3

 a4

a v1 5   a6  v v 2 5  a  7

a8   v4 v3

This is only a subgraph and not a lattice. This has subgraphs which are lattices as well as graphs.

Consider, v1  v v 2 P1 = 5  

 v3  v4

P1 is a subgraph of S1.

a4 P2 = 

a5   a6

 a7

 a8

P2 is a subgraph which is a distributive lattice.

136 Pseudo Lattice Graphs and their Applications to Fuzzy…

a1 

P3 = a2   a3

 a 4

is a subgraph which is a sublattice also Boolean algebra of order 22.

v5 P4 =  v1 v4  

 v2=a7  v3  a8

is a subgraphs which is not a lattice.

All the subgraphs P1, P2, P3 and P4 are connected subgraphs of the pseudo lattice graph of type II.

However all lattices are always connected graphs.

We also have subgraphs which are not connected which are as follows:

v5  T1 =  a8

v4   v2 =a7

 v  3

The above subgraph T1 is not connected.

Pseudo Lattice Graphs of Type II 137

a1 

T2 = a2   a3

 a4

v1 

v5   v3

 v4

The subgraph T2 is also not connected.

In view of all these we have the following theorem.

THEOREM 3.2: Let PGL = {Collection of all pseudo lattice graphs of type II merging vertices / edges of the graph G and lattice L}.

(i) G is a subgraph of every P in PGL. (ii) L is a subgraph which is a lattice in every P in PGL. (iii) Every subgraph of PGL need not in general be a connected graph. (iv) P  PGL has connected subgraphs.

The proof is direct and hence left as an exercise to the reader. 138 Pseudo Lattice Graphs and their Applications to Fuzzy…

a1  Example 3.18: Let L =   a  a3 a4 2 a5

and v1  v2 v5   G =

 v3  v4

be a lattice and a graph respectively.

v 1  v2 v5   P =

 v3=a1  v4   a  a3 a4 2 a5

This P is a pseudo lattice graph of type II which is only a graph.

This has subgraphs which are lattices as well as subgraphs some are connected subgraphs. Some disconnected subgraphs of P exist.

For all vertices alone is a subgraph of P which is totally disconnected.

Pseudo Lattice Graphs of Type II 139

v1 v2  

  a1=v3 v4    

a2 a3 a4

is a subgraph of P which is disconnected.

v1 v2  

v4   v3=a1 

v5  a3

is again subgraph which is disconnected. Now consider the pseudo subgraph is as follows.

v1 

v2 v5 

 v3=a1  

a2  a3 a4

a5 140 Pseudo Lattice Graphs and their Applications to Fuzzy…

This is only a subgraph and not a lattice.

We can have connected and not connected subgraphs of P.

Example 3.19: Let

L =  a1

v1   a2

  a3 v6  v2

 a4

v5   v3 a5   a6

 

v4 a7

be a lattice and a graph. We can get several such pseudo lattice graphs of type II.

We get a pseudo lattice graph of type II.  a1 v1  v  2

v6   v2=a3

v   v3=a4 5  v4=a5  a6

 a7

This is only a graph and not a lattice. Pseudo Lattice Graphs of Type II 141

 a1

 a2 S1 =

v6   a3=v2

 v3=a4 is a graph and not a lattice.

Consider

v3=a1 

v4=a5   a 6  a7 a subgraph which is lattice infact a Boolean algebra of order four.

We have seen subgraphs of the pseudo lattice graphs of type II.

Consider the following example.

Example 3.16: Let

a1 L1 = 

a2   a3

a5   a4  a 6 142 Pseudo Lattice Graphs and their Applications to Fuzzy…

and  v1 v2 

v6  G =  v3  v7  v5  v4

be a lattice and graph respectively.

We can have several pseudo lattice graphs of type II got by merging vertices or edges or both.

a1  a3 =v2  v1 a2  P1 = 

a5  v6=a4   v3   a6  v5  v 7 v4

This is a pseudo lattice graph of type II which is only a graph and not a lattice.

Consider P = a1 2 

a2   a3=v3

a5  v6    a4 v4   a6  v5 v7 Pseudo Lattice Graphs of Type II 143

The pseudo lattice graph P2 of type II is only a graph different from P1.

a1 

a2   a3 P3 = a5   a4  v1=a6

v6   v3 v2  v4 v6   v 3

v7 

The pseudo graph of type II; the resultant is only a graph and not a lattice different from P1 and P2.

We have several subgraphs.

 a1

 a 2 a5 

a5   a 2 a5  is a subgraph. 144 Pseudo Lattice Graphs and their Applications to Fuzzy…

 a1 a2 

a4   a3

v6   v5

 v7

a1 a3 a2 a4 v6 v5 and v7 are the vertices of the subgraph.

 a 3  a4 v6=v1  v   3 v2

 v4

is again a subgraph.

Example 3.21: Let

G = v1 

v4   v3  v2

v5   v6

 v7

v9   v8 v 11  v 10 v13 v12 Pseudo Lattice Graphs of Type II 145

and L =

a1 

a2    a a3 4

 a5

a6   a7

 a 9

 a10

be the graph and lattice respectively. We can get several pseudo lattice graphs of type II using them. They are as follows:

v1  v4   v 3  v2 v5  v6

 v7 v9  v8

v11=a1  v10

a2=v13   a  v12=a4 3  a5

a6   a7

 a9

 a10

146 Pseudo Lattice Graphs and their Applications to Fuzzy…

We see this pseudo lattice graph of type II is not a lattice.

a1=v11 

v13=a2   a  v12=a4 3

a5

is a subgraph which is sublattice.

v1 W = 

v4   v3  v2

v5  v6

is a subgraph which is a tree.

 v3

 v7 = B

v9  v8

v10 Pseudo Lattice Graphs of Type II 147

v11=a1 

v13=a2   a3

 a 5

a7  a9

 a10

B is a subgraph which is not a connected subgraph.

We can consider number of lattices and graphs (say t lattices t  1 and n  t number of graphs) and merge n of the vertices or n of the edges or merge say some r of the vertices so that all of them are merged in some way or other, that is the merging is done in such a way that no lattice lattice or graph is left out, without being merged with another graph so that an unbroken cycle is set.

We will illustrate this situation by the following examples. We call the resultant graph as the pseudo lattice graph of type II.

Example 3.22: Let L1, L2, L3 and G1, G2 be the lattices and graphs which are as follows:

148 Pseudo Lattice Graphs and their Applications to Fuzzy…

a1  L1 =

a2   a3

 a4

a5   a 6

 a7

 c1  b1 L2 = and L3 = c2   b2  c4  b3 c3   b4  c5  b5  b6

v1 u1   G1 = and G2 = v2   v4 u2   u3

v 3

We give a pseudo lattice graph of type II which is as follows:

Pseudo Lattice Graphs of Type II 149

 b1 u1 c1    b2   b3=c4 c1

 b4 c3=a1 c5     b5 u3  b6=v1 a2   a 3

v2   v4  a4

 a5   v3 a6

 a7

We see B is a pseudo lattice graph of type II. This is only a graph and not a lattice.

Consider the following graph S.

 c1 u1  c2  a1   u2=c4   b1 u3  c3 a2   a3=b2  v1  a4

v2   v4 a5   a6=b3 v3  b4  a7  b5

 b6 150 Pseudo Lattice Graphs and their Applications to Fuzzy…

Clearly S is not a pseudo lattice graph of type II for they are in two disjoint representation where only one of them is a pseudo lattice graph of type II and another is only a pseudo lattice graph of type I.

Thus we see every lattice or graph should be merged so that they are not two separate entitles.

Thus there is atleast one graph or lattice which is merged with more than one graph or lattice.

Unless this is done the resultant graph is not a pseudo lattice graph of type II.

v2 Example 3.19: Let   v4 v1 G1 =   v3

v5

w1  u2  u3 u1 w2   w5  w3 G =  and 2 w4 w6 w7  u4  u5

be 3 graphs and

Pseudo Lattice Graphs of Type II 151

a1 L1 =  a a 3 2    a4

   a7 a6 a5

 a8

be a lattice. We have the following pseudo lattice graphs of type II.

w1  u2  w2 u2=w3 u 1    w5    w4 w6 w7 v1 = u5  u4= a1  a a 3 2    a4 v2     a7 a6 a5

 v3  a8  v5

This a pseudo lattice graph of type II which is a graph. 152 Pseudo Lattice Graphs and their Applications to Fuzzy…

The graph G2 is merged with all the graphs and lattice, however the graph G3 is merged only with G2 however G2 is merged with G1.

Example 3.20: Let

 a1  b1 a  L1 = 2   a 3 and L2 = b2   b3 a4  b4  a 5

be two lattices.

w1 v1  

w   G1 = and G2 = 2 w4

 v3   v2 w3

  w5 w6

be two graphs.

Using the method of merging of the vertices and or edges we get a pseudo lattice graph of type II. Pseudo Lattice Graphs of Type II 153

 b1  a1  v1=b2 a2   a 3  a4=w1

v3   v2=b3 w   2 w4  b4 a =w 5 3

w  w 5 6

Clearly the resultant is not a pseudo lattice graph of type II.

Infact we have only two pseudo lattice graphs of type I for we see they are not merged as per the definition of pseudo lattice graph of type II.

This example is mainly to show that it is mandatory for all the lattices and graphs to be merged with each other so that it is not like the Example 3.24.

Consider the merging of vertices b1 with a5, v2 and w5.

We get a pseudo lattice graph of type II which is as follows.

154 Pseudo Lattice Graphs and their Applications to Fuzzy…

w1 S =   a1 a3 w2  w  a2  4  v1

w3  a4

a5=b1=w5=v2 w5    v3

 b2

 b3  b4

This is a graph and when a vertex of every graph is merged we call such pseudo lattice graphs of type II as strongly merged graph.

If that point or vertex is removed we call the resultant graph as the dismantled graph.

For we see if the vertex b1 (a5, v2 and w5) is removed the resultant is four disjoint or non connected subgraphs which is as follows:

w1 v1  a1  

w2   w4 v3  a2  a3

  w3  a 4  w6 Pseudo Lattice Graphs of Type II 155

 b1  b2

 b3

Thus we see in the subgraphs, three are lattices and one is a semilattice. Such merging (or bonding of a single vertex is a strong vertex merge, but one can easily dismantle that graph also. We can also get the strongly edge merged graphs which is as follows:

a1 

a2  a 3  a4=v1=b1=w1

v3 w2    a5=v2=b2=w4  b  3 w3  b4

 w6

The resultant pseudo lattice graph of type II is a strongly edge merged pseudo lattice graph of type II. The removal of that edge dismantles the graph leading to four subgraphs.

 b3

a1  b4 

a  2  a3 w3  w2  v3 

w  6  w5 156 Pseudo Lattice Graphs and their Applications to Fuzzy…

 a1

a 2   a3

 a4

 a5

removed of edge a4a5 results in

v1 a1  

a2    v3 a3

 v2

Removal of edge v1v2 results in  v 3 Removal of edge w1w4 results in

 w3 w2 

w6   w5

Removal of edge b1 b2 results in

 b1

 b2  b3

  b4 b3  b4 Pseudo Lattice Graphs of Type II 157

Thus we get the dismantled subgraph which is entirely different. We see strong merging by vertices or edges becomes dismantled if that vertex or edge is moved.

If in case of n1 lattices and m1 graphs is a weakly merged pseudo lattice graph if we have maximum number of merging vertices or edges is two that for a lattice or graph is merged to a maximum of two lattice or graph.

This will be illustrated by the following examples.

Example 3.25: Let b1  a1  b2 L1 = L2 =   b3 a3 a2     a4 b4  a    b5 7 a6 a5  b6  a 8

v  c 1 L3 = 1 G1 = 

 c2 v   2 v5  c3 v3  v4 c 4   c5  c6 v6  c7 v   v  7 8 c8

158 Pseudo Lattice Graphs and their Applications to Fuzzy…

and w1  w2 w5    w6 G =  w3 2  w  4 w8  w7

the lattices and graphs. We get the following pseudo lattice graphs.

a1  v1  a3 a2    b1 a4  v2   v5

a    b3= v3   7 a6 v4  c b2=a5 1   b4 v6  c2 a8 w1=v8  b5  c3 v7   w2 w5   b6  c4  c5  c6  w3  w6 c7  w  4 w8  c 8  w7

We see each of the lattices or graphs are maximum merged in twos.

The subgraphs are as follows:

Pseudo Lattice Graphs of Type II 159

a1 

b1 v2   a2   a6    v4 a7=b2 b3=v3  a8

w1 c3   w2 w5   c6   c5  c6

 w3 c7  w4  c8  w 7

We can have subgraphs of order one  w 4 viz. one vertex, subgraphs of order two viz. two vertices or an edge connecting the vertices.

v1   v6

We can have three vertices or three vertices and an edge or three vertices and two edges.

c3 w1   and c1 b1 w2    c4  b w3  2  b3 

We work with subgraphs and get subgraphs of very many different orders. 160 Pseudo Lattice Graphs and their Applications to Fuzzy…

We can have order four subgraphs which are as follows:

b1 w1  b3   w2 w4   b2  b4 

w3 

 v 5 a1   v6     a2 a3 v7 v8

 c3 a4 

a6   a7    c6 c5 c 4  a8

and so on.

These merging of graphs can play a vital role while working with merging of a node or concept in FCMs (Fuzzy Cognitive Maps) model. So that the merging concept will give a larger dynamical system. Likewise merging of a graph with two vertices and an edge will result in a some way or other a partially combined FCMs. So one can think of getting more and more FCMs models by this method. Pseudo Lattice Graphs of Type II 161

Thus at this juncture we see this merging of graphs also has more applications in both FCMs model, FRMs (Fuzzy Relational Maps) model and FREs (Fuzzy Relational Equations) models.

Since we can with out loss of generality assume all lattices are trivially or obviously graphs we have no problem in merging a graph with a graph by a vertex or an edge or both or by collection of vertices or several vertices and edges.

Let us consider two directed graphs.

 w1 v1  x  y1 1 

 w2 v2   y2 x2   w3  y v3  3 x3   w4  y4 x4   y5

We see the two graphs are merged in this manner wi is merged with xi, i = 1, 2, 3, 4.

w1=x1

  y1 v1  w2=x2  y  2 v2  w3=x3  y3  v3  w4=x4   y4

 y5 162 Pseudo Lattice Graphs and their Applications to Fuzzy…

This pseudo lattice graph of type II will also be known as the linked graph.

This is the type of graphs associated with FRMs or FREs.

Likewise we can merge one vertex or an edge or more vertex and get a merged FCM. Both these concepts will be defined and developed in this chapter.

Let us suppose we have two experts working on the same problem.

However both the experts work with a different set of concepts but they have some nodes to be in common. In regards of some common nodes / concepts they have some edges also to be common.

Now if the two direct graphs G1 and G2 be given by the two experts. We can take the directed graph of the experts and merge the common nodes / edges get a new graph the pseudo graph and now using this pseudo graph we can analyse the problem.

We define the FCMs which has merged directed graphs will be defined as the merged or glued FCMs. Such study is interesting and leads to many results in FCM models as they are not combined FCM but some what merged FCMs.

This model will be illustrated by the following examples.

Example 3.26: Let us consider a study of any nations political situation, that is the prediction of electoral winner or how people tend to prefer a particular politician and so Pseudo Lattice Graphs of Type II 163

on and so forth involves not only a lot of uncertainly for this no data is available.

They form an unsupervised data. Hence we are at the outset justified in using FCM model.

Suppose it is from India the Indian politics is analysed using six nodes.

x1 - Languages x2 - Community x3 - Service to people public figure configuration and personality ad nature x4 - Finance and media x5 - Party’s strength and opponents strength x6 - Working member for the party.

Suppose we have two experts. Experts one E1 uses the four nodes x1, x2, x3 and x5 and expert to E2 uses the four nodes x3, x4, x5 and x6. The directed graph given by experts E1 one is

1 x1 x2

1 1 1

x3 x5 1

The directed graph given by expert E2 is as follows.

164 Pseudo Lattice Graphs and their Applications to Fuzzy…

1 x x 3 4

1 1 1 1 1

x5 x6

The merged two edges x3 x5 of E1 and E2 is as follows:

1 1 x4 x1 x3

1 1 1 1 1 1

x6 x2 x5 1

Now the above graph is a merged graph.

The connection matrix of the MFCM (Merged Fuzzy Cognitive Maps) is as follows.

xxxxxx123456

x1 011000   x2 101000 M = x 000110 . 3   x4 001011 x 111000 5   x6 001000

Pseudo Lattice Graphs of Type II 165

Now using this merged FCM we are in a position to get the merged opinion of the two experts. This is not the combined opinion only a merged opinion.

We can use these and get the merged opinion of both the experts. This saves times and also gives equal importance to both the experts.

We will give one more example of them.

Example 3.27: Let X1, X2, …, X6, X7, X8 and X9 be seven attributes / nodes associated with the problem.

Let the first expert works with the nodes X1, X2, X8, X5, X6 and the second expert works with the second expert nodes X1, X5, X4, X3 and X7.

The directed graph given by the

x2

x1 x8

x6 x5

First expert using the nodes X1, X2, X9, X8 and X6.

The directed graph using the nodes X1, X5, X4, X3 and X7 is given by the second expert is as follows:

166 Pseudo Lattice Graphs and their Applications to Fuzzy…

x3

x1 x4

x5 x7

We now merge the vertex X1 of the two graphs.

x2 x 3 x8 x1 x4

x6 x9 x5 x7

So that we get the over all model using all the nine nodes.

Thus by this method we get the Merged FCM (MFCM).

Such applications are very useful in the study of fuzzy models.

We now show by examples how merging of graphs give new models in case of FRM and FRE. Pseudo Lattice Graphs of Type II 167

Example 3.28: Let us consider FRM given by two experts working with only one common set of concepts; how to relate by merging them.

Let us consider three sets of attributes S1, S2, S3, …, S5 and R1, R2, R3, R4 used by expert one and R1, R3, R5, R6 and T1, T2, T3, T4, T5 and T6 are the attributes worked by the second expert.

We give the Fuzzy Relational Maps (FRMs) directed graph of the first expert is as follows:

S1   R1

S2   R2

S3   R3

S4   R4

S5 

The directed graph of the second expert is as follows:

 T1 R1 

 T 2 R3 

 T3

R5   T4

R6   T5

 T6 168 Pseudo Lattice Graphs and their Applications to Fuzzy…

We can now get the second expert opinion.

S1   T1 R1   T2 S2  R2   T3 S3  R3   T4 S4  R4   T5 S5  R5   T6 R6  We get the following directed graph.

 T 1 S1 

 T2 S2   T 3 S3   T4

S4   T5

S5   T6

This is the way merging results in a new graph and hence new FRMs. On similar lines we can have FREs whose bigraphs can be merged at one or more vertices.

Let us consider X1, X2, …, X7 some 7 concepts related with a problem Y1, Y2, …, Y5 be some five concepts related with the problem. If X1, X2, …, X7 is taken as the domain space and Y1, Y2, …, Y5 as the range space of the Pseudo Lattice Graphs of Type II 169

FRM then we get the following bigraph related with the FRMs.

X1   Y1 X2   Y2 X3   Y3 X4   Y4 X5   Y 5 X6 

X7 

Suppose another expert works with Y1, Y2, …, Y5 as the domain space and say Z1, Z2, …, Z6 as the range space we get the following bigraph

 Z 1 Y1 

 Z2 Y 2   Z 3 Y3   Z4

Y4   Z5

Y5   Z6

170 Pseudo Lattice Graphs and their Applications to Fuzzy…

Now be merge the two bigraphs on the five vertices we get the pseudo graph which is as follows.

X1   Z1  Y1 X2   Z2  Y X3  2  Z3  Y3  Z X4  4 

Y4  Z5 X5   X6  Y5  Z6

X7 

X1 

 Z1 X2   Z2 X3   Z3 X4   Z4

X5   Z5 X6   Z6 X7 

Thus merging FRMs leads to or works like linked FRMs. On similar lines we can use the bigraphs of FREs and merge them to get a bigraph. Pseudo Lattice Graphs of Type II 171

Thus merging of vertices or edges graphs of fuzzy models gives us the merged fuzzy model.

This newly constructed merged fuzzy model like merged Fuzzy Cognitive Maps, merged Fuzzy Relational Maps and merged Fuzzy Relational Equation play a vital role in studying social problems in studying social problems and interlinking or merging the attributes resulting new results.

Thus we can using the concept of merging of graphs construct new merged fuzzy models.

Infact we can also merge more than 3 graphs of 3 fuzzy models working on the same problem and get a new merged model and so on and so forth

C1 C2

C4

C3

C5

172 Pseudo Lattice Graphs and their Applications to Fuzzy…

C6 and

C1 C 9 C10 C9

C8 C11

be three directed graphs given by three different experts.

We see the connection matrices H given by the three graphs are as follows.

ccccc12345

c1 01000   c2 00110 E1 = c 00001 3   c4 10000   c5 10000

cc16 cc 89

c1 0110   E2 = c6 0011 c 0000 8   c9 0000

Pseudo Lattice Graphs of Type II 173

cc91011 c

c9  001 and E3 =   c00010     c11011  

We see the graphs of the first and second expert have the vertex C1 to be the common vertex to be merged.

For the expert two and three C9 is the common vertex which is to be merged.

C6

C 1 C2

C8 C 4

C3

C C10 9

C5 C 11

Now we using this merged graph obtain the connection matrix of the merged model. 174 Pseudo Lattice Graphs and their Applications to Fuzzy…

cccccccccc123456891011

c1 0100011000  c2 0011000000 c 0000100000 3  c4 1000000000 E = c 1000000000 5  c6 0000001100 c 0000000000 8  c9 0000000001  c10 0000000000  c11 0000000100

Now E1 is the merged FCM and the consolidated one will give the opinion of all the three experts.

However it is distinctly different from the combined FCM.

Thus this new merged model can at a time give the hidden pattern in a consolidated way their by saving time and economy.

We will some more illustrations of them.

Suppose one works with a problems with nodes C1, C2, …, C10. Three experts work on the problem and two of them have the node C1 and C2 in common and other two them have the node C7 and C8 in common. The directed graph given by the three experts are as follows.

Pseudo Lattice Graphs of Type II 175

C2

C1

C9

C4

C5

The above is the directed graph given by the first expert. The directed graph given by the second expert is as follows.

C2 C1

C7

C8

C 6

The direct graph given by the third expert is as follows:

176 Pseudo Lattice Graphs and their Applications to Fuzzy…

C 7

C 8 C 3

C10

The connection matrices of the following three directed graphs E1, E2 and E3 are as follows:

ccccc12459

c1 01010   c2 00101 E1 = c 00000 4   c5 00101   c9 00000

ccccc12678

c1 01010   c2 00101 E2 = c 00000 6   c7 00000   c8 00010

Pseudo Lattice Graphs of Type II 177

cc37 cc 810

c3 0010   and E3 = c7 1010 c 0001 8   c10 1000

We get the merged graph which is as follows:

C10 C3

C 7 C8

C2

C1

C6 C9

C 4

C5

Let E be the merged connection matrix of the merged graph which is as follows:

178 Pseudo Lattice Graphs and their Applications to Fuzzy…

cccccccccc12345678910

c1 0100101000  c2 0001010110 c 0000000100 3  c4 0000000000 E = c 0001000010 5  c6 0000000000 c 0010000100 7  c8 0000001000  c9 0000000000  c10 0010000000

Using this matrix E as the merged dynamical system one can work with the fuzzy models.

In the same way merged FRMs and merged FREs are constructed. Thus the merged graphs play a vital role in this study.

Chapter Four

PSEUDO NEUTROSOPHIC LATTICE GRAPHS OF TYPE I AND TYPE II

In this chapter we define the concept of pseudo neutrosophic lattice graphs of type I using two lattices in which atleast one should be a neutrosophic lattice. We also define pseudo neutrosophic lattice graph of type II in which atleast one of the lattice or the graph must be neutrosophic.

In the case of type II we also make use of both graphs where atleast one of them is a neutrosophic graph. Finally we give the applications of these pseudo neutrosophic lattice graphs of type II when two graphs are used in fuzzy neutrosophic models. These new fuzzy neutrosophic models are termed as merged fuzzy neutrosophic models.

For definition of neutrosophic graphs refer [79, 89]. For the concept of lattices and neutrosophic lattices refer [87].

180 Pseudo Lattice Graphs and their Applications to Fuzzy…

DEFINITION 4.1: Let L1 and L2 we any two neutrosophic lattices we can merge the vertices or edges or both and get a pseudo neutrosophic lattice of type I. This can be extended to any number of neutrosophic lattices L1, L2, …, Ln; n < .

It is pertinent to keep on record that all lattice need not be neutrosophic but atleast one lattice must be a neutrosophic lattice.

We will first illustrate this by some examples.

a1 Example 4.1: Let L1 = and L2 = I  a2  a  3 aI  a  a4   a5 0

 a6 be two lattices a neutrosophic lattice L2 and a lattice L1. We can merge vertices of L2 with any of the vertices of L1.

The resultant graph is defined as the neutrosophic pseudo lattice graph of type I.

More merging of vertices is possible only when we take both the lattices to be neutrosophic lattices.

Example 4.2: Let L = 1+I 1 

a1I   a1

  a2I a2

 0 Pseudo Neutrosophic Lattice Graphs of Type I and II 181

1+I



b and L2 = b1I  1  0 be any two neutrosophic lattices. We can merge 1 + I with 1 + I and zero with zero and rest no other merging.

1+I 

a I 1   a1 b1I b1 

a2I   a2

 0 be the neutrosophic pseudo lattice graph of type I.

1+I  I  a1=b1 b I a1I    1   a2I  a2 0

 0 This is a neutrosophic pseudo lattice graph of type I which is only a neutrosophic graph.

182 Pseudo Lattice Graphs and their Applications to Fuzzy…

Example 4.3: We can also have neutrosophic lattices with both vertices and edges to be neutrosophic. Then we can have merging of the real edges or merging of the neutrosophic edges resulting in pseudo neutrosophic lattices.

Let a1 a2  

a5   a4

 a5

be the edge neutrosophic lattice L1 and

b1  b2  b3

 b4 I  b5 L2 =

b6  b7  b8

be any edge neutrosophic lattice.

We can merge edge a1a4 with b2b3 and obtain the pseudo neutrosophic lattice graph which is as follows: Pseudo Neutrosophic Lattice Graphs of Type I and II 183

a1 a2  

a5   a4

 a5  b3

b4 b5 

b6  b7  b8

The resultant is only a neutrosophic graph. We have merged the neutrosophic edge with a neutrosophic edge.

We can also merge b5b6 with a4a3 and get a pseudo neutrosophic lattice graph which is as follows:

b 1 a1 a2  b2    b3  b4 a5  I  b5=a4

b a3=b6   7  b8 This is also a pseudo neutrosophic lattice graph which is only a neutrosophic graph which is not a lattice. We can get several such pseudo neutrosophic lattice graphs. 184 Pseudo Lattice Graphs and their Applications to Fuzzy…

b1

 b2

b  3  b4  b5 a a1 2 b6  b7 

 b8  a 4

 a 5 This is again a pseudo neutrosophic lattice graph of type I which is also a neutrosophic lattice.

We can get this type of pseudo neutrosophic lattice graphs of type I.

b1

 b2

 b3

 b4 a1 I a2   b5

b6  b7 a 5    a4 b 8  a5

This is a yet another pseudo neutrosophic lattice graph of type I by merging the vertices a2 with b5.

Pseudo Neutrosophic Lattice Graphs of Type I and II 185

Example 4.4: Let us consider the following two neutrosophic lattices L1 and L2.

a1   b0

 b1  a2   a3 a5  a6 b  b3 2

 b4  a7  b 5  a8

 a9 We can merge the neutrosophic edges a8a9 with b0b1 and get the following pseudo neutrosophic lattice graphs of type I.

a1 

a  a3 a5   a6 2

 a7

 a8=b0

 b1=a9

b b2   3

 b4

 b5

This is again a neutrosophic lattice. 186 Pseudo Lattice Graphs and their Applications to Fuzzy…

We can merge the edges b1b2 with edge a1a2 and get the pseudo neutrosophic lattice graph of type I which is as follows:

 b0

 a1=b1

a   b  b3 a2  a3 5 a62

 b4

  b5 a7

 a8

 a9 We see it is not a neutrosophic lattice which is also a neutrosophic graph.

Example 4.5: Let us consider the following three neutrosophic lattices

 b1

L1 =  a 1 L2 =  b2 a a2  3 b b3  4

 a4  b5  a5

and L3 =  c 1 .  c2

 c3

 c4

Pseudo Neutrosophic Lattice Graphs of Type I and II 187

We now merge edges a4a5 with b2b5 and merge vertex b 5 with c1.

 a1

a2  a3  b2   b5  b4 b3 a  5  b5=c1

 c2

 c3

 c4 The neutrosophic pseudo lattice graph of type I is only a neutrosophic graph.

Thus we can merge more number of neutrosophic lattices and obtain a pseudo neutrosophic lattice graph of type I. Interested reader can construct more of them we can also as in case of usual pseudo lattice graphs of type I find substructure in case of neutrosophic pseudo lattice graphs of type I which will be illustrated in an example or two.

a1 Example 4.6: Let L1 =  a2  a3 b and L2 =  1  a4

b3  a5  b2  a6 b5   b4  b6 188 Pseudo Lattice Graphs and their Applications to Fuzzy…

be the two neutrosophic lattices. We can merge edge a3a4 with b1b4 and get the neutrosophic pseudo lattice graph which is as follows.

a1  a2  a3=b1 b3  b2

b5   a4=b4   a5 b 6  a6

Clearly the neutrosophic pseudo lattice graph of type I is not a neutrosophic lattice only a neutrosophic graph.,

Consider

b  1 b3 

P1 = b5  b4=a4

 a5 b6 a6

P1 is a neutrosophic subgraph of type I.

Pseudo Neutrosophic Lattice Graphs of Type I and II 189

a =b 3 1

b2 P2 = b3    a4 = b4

P2 is the neutrosophic subgraph which is also a neutrosophic lattice.

a3=b1 Consider  a1  a5 P3 =  b4 = a4 b5   a  2 b6 be the subgraph which is only a subgraph and not a sublattice.

We can have several such subgraphs. We see P1, P2 and P3 happen to be neutrosophic subgraphs.

We can also have subgraphs which are not neutrosophic for

a3=b1 

P4 = b3  b2

 a4 = b4

P4 is a subgraph which is not neutrosophic, it is also a sublattice which is not neutrosophic. 190 Pseudo Lattice Graphs and their Applications to Fuzzy…

Thus we see in general a neutrosophic pseudo graph lattice can have a subgraph which is a neutrosophic lattice or which is not a neutrosophic lattice or a graph which is a neutrosophic graph or not a neutrosophic graph so this neutrosophic lattice graph of type I can have four types of substructures.

Example 4.7: Let L1 and L2 be any two neutrosophic lattices which is as follows:

a1 L1 =  and L2 =

 b1 a2   a 3  b2 b4 a6   a4 b3    b5 a5

We can merge edges a4a5 with b1b2 and b2b3 with a5a6.

We get S the following neutrosophic pseudo lattice graph of type I.

a1 

a2   a 4

a3   a =b 5 2 b4  a6=b3  b5

Pseudo Neutrosophic Lattice Graphs of Type I and II 191

We see the resultant is a neutrosophic lattice.

This S has all the four types of subgraphs which are described in the following:

a1 

a 2   a4=b1 P1 =

  a3 a5=b2

is a subgraph which is not a neutrosophic subgraph.

a5 = b2  P = 2 b a6=b3  4  b5 P2 is a subgraph which is a neutrosophic sublattice of S.

a1 

a 2   a4 P3 =

a3   a 5  b  4 a6=b3

P3 is a subgraph which is a neutrosophic subgraph of S.

192 Pseudo Lattice Graphs and their Applications to Fuzzy…

a1 

 a 2  a3

= P4. P4 is a subgraph which is a lattice which is not neutrosophic. We can also have cranky or unnatural merging in neutrosophic graphs. That is we try to merge a neutrosophic vertex with a non neutrosophic vertex or a neutrosophic edge with a non neutrosophic edge.

We call the merged neutrosophic lattice as neutrosophic cranky pseudo lattice graphs.

We will give examples of cranky neutrosophic pseudo lattice graphs.

Example 4.8: Let L1 =

 a1

 a2

 a3

a a5  a6 4   a7 and L2 =

 b1

 b2 b b6 b3  b4  5 

 b 7  b8 Pseudo Neutrosophic Lattice Graphs of Type I and II 193

be any two neutrosophic lattices. We merge edge a1a2 of L1 with the neutrosophic edge of b7b8. Let S be the resultant cranky neutrosophic pseudo lattice graph of type I whose graph is as follows:  b1  b2 b5 b6 b3 b4 

 b7=a1

 b8=a2

 a3 a a4 a5  6  a7

S is a neutrosophic lattice.

Now when a neutrosophic edge is merged with the real edge we always make it only as a neutrosophic edge. This is the assumption or definition made in this book.

Interested reader can give more examples of such cranky pseudo neutrosophic lattice graphs. However we leave the following theorem for the reader.

THEOREM 4.1: Let L1 and L2 be two neutrosophic lattices. A cranky pseudo neutrosophic lattice graph of type I of L1 and L2 got by merging a neutrosophic edge or vertex with a real edge or vertex respectively is always a neutrosophic lattice graph of type I.

Now we proceed onto define neutrosophic pseudo lattice graph of type II. 194 Pseudo Lattice Graphs and their Applications to Fuzzy…

Let G be a neutrosophic graph and L be a neutrosophic lattice or one of G or L alone is neutrosophic then if we merge the edges of them or merge the vertices of them we define the resultant graph to be a pseudo neutrosophic lattice graph of type II.

We will illustrate this situation by some examples.

a1 Example 4.9: Let L1 =  a a5 a2 a3  4 

 a6

v1 G =  v2 v5  

 v3  v4

be the neutrosophic lattice and G be the neutrosophic graph. Let us merge edge v1 and a6; we obtain the pseudo neutrosophic lattice graph of type II which is as follows:

a1  P1 = a4 a5 a2 a3  v1

 a6 v2 v5  

 v3  v4

Pseudo Neutrosophic Lattice Graphs of Type I and II 195

P1 is a graph and not a lattice.

Consider a1  v1=a4 a5 a2  a3   v2  P2 = 

a6=v5  v3  v4

Clearly P2 the pseudo neutrosophic lattice graph of type II which is not a neutrosophic lattice which is not a neutrosophic graph. Now we will find subgraphs of P1 and P2 in the following

a1 

S1 = a3

 a6

v3 is a sublattice of P1 which is neutrosophic.

S2 = a4

v1

v  5 v2

v4 196 Pseudo Lattice Graphs and their Applications to Fuzzy…

is a subgraph which is neutrosophic and is only a subgraph not a sublattice.

a1 S3 = 

a  a5 2  a6

is a subgraph which is a sublattice which is not neutrosophic.

a6 = v1  S4 = v2 

 v3  v4

is a subgraph which is not a neutrosophic subgraph and is not a lattice.

a1 Example 4.10: Let L = 

a2   a3

 a4

 a5

a6   a7

 a8

Pseudo Neutrosophic Lattice Graphs of Type I and II 197

be the neutrosophic lattice and G = v1 

v2 v3  

v v v9   10 v4   6

   v5 v11 v12   v 7 v8 be the neutrosophic graph.

Suppose we merge a1 with v3 edge v3v4 with a1a2, edge v3v6 with a1a3 then we get the following pseudo neutrosophic lattice graph of type II say S.

v1 

v2 v3=a1   a2=v4 v9   v10   a3=v6

v5  a4    v11 v12  v7 v 8 a5

a6   a7

 a8

198 Pseudo Lattice Graphs and their Applications to Fuzzy…

Clearly this is not a neutrosophic lattice but only a neutrosophic graph.

We can have sublattices, subgraphs, neutrosophic sublattices and neutrosophic subgraphs which are as follows:

v1 

v3 = a1 P1 = v2  

a =v a2=v4   3 6

is a subgraph of S which is not neutrosophic.

v1  v2  P2 =

v9   v10

  v11 v12

is a subgraph which is a neutrosophic subgraph of S.

Consider  a4

 a5

a   a7 6  a8 is a subgraph which is a neutrosophic lattice. Pseudo Neutrosophic Lattice Graphs of Type I and II 199

 v1 = a1

a =v  2 4 is a subgraph which is a non neutrosophic sublattice of order two.

Example 4.11: Let L =

a 1  a2  a3  a4  a5

be a neutrosophic lattice.

v1  v2

G1 =

v4 v3

be a neutrosophic graph.

We can merge the edges the v1v4 to a4a5 and obtain the neutrosophic pseudo lattice graph of type II which is denoted by S.

200 Pseudo Lattice Graphs and their Applications to Fuzzy…

a1  a2

 a3

v1=a4  v2

a5=v4  v3

Clearly S is a pseudo neutrosophic graph which is not lattice.

B1 = a1  a2  a3

 a4

 a5

is a subgraph of S which is a neutrosophic lattice.

B2 = v1 v2

v4 v3

is a subgraph which is a neutrosophic subgraph and not a lattice.

In view of this we have following theorem.

Pseudo Neutrosophic Lattice Graphs of Type I and II 201

THEOREM 4.2: Let L be a neutrosophic lattice and G be a neutrosophic graph. S be the pseudo neutrosophic lattice graph of type II got by merging vertices or edges or both.

B be the cranky pseudo neutrosophic lattice graph of type II got by merging real edges with neutrosophic edges of neutrosophic vertices with real vertices.

(1) S has L to be neutrosophic sublattice and G to be a neutrosophic subgraph. S has also sublattices and subgraphs which are not neutrosophic.

(2) B has L to be a neutrosophic sublattice and G to be a neutrosophic subgraph. B has cranky subgraphs and sublattices.

The proof follows from the fact B is a cranky pseudo neutrosophic lattice so S has sublattices and subgraphs which are not neutrosophic. Hence the claim.

The rest can be proved by any interested reader.

Example 4.12: Let L =

a1 

a3  a a2  4   a5  a6

 a7 be a neutrosophic lattice.

202 Pseudo Lattice Graphs and their Applications to Fuzzy…

v 1 v  5 

v4 v6 v2     v7

 v8  v3

  v10 v9

be a neutrosophic graph.

We can merge edge a1 a3 with v8 v9 and edge a1a5 with v8v10 and get the following neutrosophic pseudo lattice graphs of type II.

v 1 v  5  v4 v6 v2   

 v 7  v 3  v8=a1

a5=v10 a2 v9=a3 a4     a6

 a7

Clearly S is not a neutrosophic lattice only a neutrosophic graph. Pseudo Neutrosophic Lattice Graphs of Type I and II 203

This has subgraphs which are neutrosophic lattices, non neutrosophic lattices, neutrosophic graphs and non neutrosophic graphs.

We see P1

v 1 v  5  v4 v6 v2     v7  v3 is a subgraph of S which is only a graph and not a neutrosophic graph.

v4 v6  

P2 =  v8=a1

v9=a3   a4  a5=v10

 a 7 is a subgraph which is a neutrosophic subgraph of S.

 v8=a1 P3 = a2  a4  a6

 a7 204 Pseudo Lattice Graphs and their Applications to Fuzzy…

P3 is a subgraph of S which is not neutrosophic.

Consider

 v8=a1

P4 = v9=a3   a4  a5=v10

 a 7

a subgraph which is a neutrosophic sublattice of S.

Interested reader can construct substructures. All the substructures P1, P2, P3 and P4 given here are connected.

Without loss of generality we can also have substructures which are not connected and they are subgraphs neutrosophic or otherwise

v1  a1 

v4 a2 T1 = v2   

 a4



v3  a 7

T1 is a subgraph of S. T1 is a subgraph which is not neutrosophic.

However T1 is not a lattice but T1 is only a subgraph which is not connected.

Pseudo Neutrosophic Lattice Graphs of Type I and II 205

Let T2 =

a5=v10  a1=v8 a6   a7 v9=a3  a 7 be a subgraph. T2 is not a lattice T2 is a subgraph of S which is neutrosophic and is not connected.

However all lattices are connected so we cannot have sublattices which are not connected neutrosophic or otherwise.

Now interested reader can study this situation.

Finally we can also merge the vertices or edges or both of neutrosophic graphs which we choose to call only as pseudo neutrosophic lattice of type II.

Now we will first illustrate this by some examples.

Example 4.13: Let

v1  v2 v5  G1 = 

 v3  v4

206 Pseudo Lattice Graphs and their Applications to Fuzzy…

and G2 =

w1 

w  2

w   w3 7

 w8  w4

   

w10 w9 w6 w5

be two neutrosophic graphs.

Let us merge vertices w6 with v1 and w5 with v2. We get the neutrosophic pseudo lattice graph of type II which is as follows:

w1 

w  2

w7  w3

 w8  w4 v =w 1 6     w5=v2 v5 w10 w9

 v3  v4 Pseudo Neutrosophic Lattice Graphs of Type I and II 207

Clearly S is a neutrosophic pseudo lattice graph of type II. Several such pseudo neutrosophic lattice graphs of type II can be got by merging vertices or edges or both.

We give a few substructures of them in the following.

P1 = w1 

w3  w2 

 w4

v1  v2 v5  

 v3  v4 is a subgraph of S which is not connected and it is a neutrosophic subgraph of S. Now consider P2 a neutrosophic subgraph.

 w4 v1=w6 P2 =   w5=v2

v5 

 v3  v4

208 Pseudo Lattice Graphs and their Applications to Fuzzy…

P2 is a subgraph of S which is not neutrosophic but is connected.

w2 P3 =  v2=w5  w7   w3 v5  w2  w8 

is a again a subgraph which is neutrosophic but is not connected.

Thus we find these pseudo neutrosophic lattice graphs of type II when two neutrosophic graphs are used find applications in neutrosophic fuzzy models like Neutrosophic Cognitive Maps (NCMs) models, Neutrosophic Relational Maps (NRMs) models and Neutrosophic Relational Equations (NREs) model as all these three models function on neutrosophic directed graphs.

These will be illustrated by the following examples.

Example 4.14: Let G1 and G2 be two neutrosophic directed graphs associated with the Neutrosophic Cognitive Maps (NCMs) model of two experts who work on the same problem.

G1 = c1   c2

 c3

c5   c4

Pseudo Neutrosophic Lattice Graphs of Type I and II 209

be the neutrosophic directed graph given by the first expert for the NCMs.

Let G2 = c1   c6

c5   c8

 c7 be the neutrosophic directed graph given by the second expert using the same NCMs node for the same problem.

Now we see both the graphs G1 and G2 have the vertices C1 and C5 in common and C1C5 edge is also common.

Thus we can merge these two directed graphs of the NCMs. By merging C1C5 of them we get the following directed neutrosophic graph S.

 c2

c1   c6

 c3

 c4 c5   c8

 c7

The neutrosophic connection matrix of the neutrosophic graph given by the first expert is as follows:

210 Pseudo Lattice Graphs and their Applications to Fuzzy…

ccc123 cc 45

c1 01011   c2 00010 M1 = c 00000 3   c4 00000   c5 0I100

The connection neutrosophic matrix of the neutrosophic directed graph given by the second expert is as follows:

ccc156 cc 78

c1 01110   c5 0000I M2 = . c 00001 6   c7 00I00   c8 00000

Now we obtain combine neutrosophic connection matrix of the pseudo neutrosophic lattice graph S of type II.

cc1234 c c c 5678 c cc

c1 01001110   c2 00010000 c 00000000 3   M = c4 00000000 . c 0I10000I 5   c6 00000001 c 00000I00 7   c8 00000000 Pseudo Neutrosophic Lattice Graphs of Type I and II 211

Now M gives the merged dynamical system of the two experts opinion of the NCMs. This new matrix functions for both the experts opinion in a merged way. This application has two advantages.

In the first place it gives equal importance to both the experts and secondly we work with the single dynamical system time which can save time and economy.

Example 4.15: Let us consider three directed neutrosophic graphs of NCMs related with the same problem.

G1 = c1   c3

c 2   c4

c5 

G2 = c1   c3

c6   c7

c6  and G3 =  c8

c9 

212 Pseudo Lattice Graphs and their Applications to Fuzzy…

We see these three graphs can only be merged in a unique way that is one and only way which is as follows:

c9 

c6   c8 S =

c1   c7

 c3

c 2   c4

c5 

We give the connection neutrosophic matrices of all the three directed graphs.

M1 is the connection neutrosophic matrix of graph G1,

ccc123 cc 45

c1 00100   c2 000I0 M1 = c 01000 3   c4 00001   c5 10000

Pseudo Neutrosophic Lattice Graphs of Type I and II 213

The neutrosophic connection matrix M2 of the graph G2 is as follows:

ccc136 c 7

c1 0100   M2 = c3 0011 c 100I 6   c7 I000

The neutrosophic connection matrix of the graph G3 is as follows:

ccc689

c0016   M3 =   . cI008     c0109  

Now we get the connection neutrosophic matrix S of the pseudo neutrosophic lattice graph of type II got after merging the edge c1c3 of G1, with the edge c1c3 of G2 and the vertex c6 of G2 with c6 of G3.

cc12345 c c c ccc 6789 c

c1 001000000   c2 000I00000 c 010000100 3   c 000010000 M = 4   . c 100000000 5   c6 100000000 c I00000000 7   c8 00000I000   c9 000001010 214 Pseudo Lattice Graphs and their Applications to Fuzzy…

Thus by working with this M as the dynamical system time is saved and all the work (that is experts opinion) is consolidated as a single system.

Thus we get the merged neutrosophic cognitive maps model which is better than the combined NCM model.

Now we describe this suppose G1, G2, …, Gn are the neutrosophic directed graphs given by n experts who work on the same problem. We have every graph has atleast a common edge or a common vertex.

Thus we merge all the n-graphs together to obtain a neutrosophic pseudo lattice graph as the merged opinions of the experts we work with the merged NCMs.

This study is new and interesting. Next we describe the merged Neutrosophic Relational Maps model and Neutrosophic Relational Equations model. Such study has been carried out in chapter III for FRMs and FREs models.

We will illustrate this situation by some examples.

Example 4.16: Let us consider the neutrosophic directed graphs H1 and H2 given by two experts studying the same problem using the Neutrosophic Relational Maps model.

D1 

 R1

D2   R2

D3  H1 =  R3

D4   R4

D5  Pseudo Neutrosophic Lattice Graphs of Type I and II 215

neutrosophic bipartite graph given by the first expert. Let

D5 

 R3 D6 

 R5 D7 

 R6 D8 

neutrosophic directed bigraph of the NRMs given by the second expert. We see in the graphs H2 and H1 only the edge D5R3 is common. By merging D5R3 of H1 and H2 we get the following neutrosophic pseudo lattice graph H of type II.

D1   R1

D2   R2

D3   R3

D4   R4 D5 

 R5 D6 

 R6 D7 

D8 

216 Pseudo Lattice Graphs and their Applications to Fuzzy…

H is the merged neutrosophic directed graph of the NRM. However H is not a linked merged NRM.

Example 4.17: Let H1 and H2 be any two neutrosophic directed graphs of a NRMs given by two experts on the same problem.

D1 

H1 =  R1

D2 

 R2

D3   R3

D4   R4

D5 

 R5 D6 

and

D6 

H2 =  R2

D7 

 R3

D8   R6

D9   R7

D10

 R8 D11

Pseudo Neutrosophic Lattice Graphs of Type I and II 217

We see the two graphs can only be merged in a unique way to get at the merged NRM. The merged graph is as follows:

D1 

H =  R1

D2   R2

D3   R3

D4   R4

D5 

 R 5 D6 

D7 

 R6 D8 

D9   R7 D10

 R8 D11

Using this directed graph H we can obtain the merged connection matrix of the merged graph H.

Using H analysis of the problem can be made.

Interested reader can study such merged NRMs model. The main advantage is it saves time and economy we can also obtain a merged NRMs models using three experts or more. Just for the sake of the simplicity we give an 218 Pseudo Lattice Graphs and their Applications to Fuzzy…

example where 3 experts opinion on an NRM model; say the graphs of the NRM model given by the three experts be G1, G2 and G3 where

D1  G1 =  R1

D2   R2

D3   R3 D4   R4 D5 

G2 = D5   R3

D6   R4

 R5 D7 

 R6

D7  G3 =  R3 D8   R5 D9   R7 D10  R8 D11  R 9 Pseudo Neutrosophic Lattice Graphs of Type I and II 219

Now we find the merged neutrosophic graph (that the pseudo neutrosophic lattice graph of type II. We see for the neutrosophic graphs G1 and G2 the edge D5  R3 and the neutrosophic edge D5R4 are to be merged and for the neutrosophic graphs G2 and G3 the edge D7R3 are merged. Now we have H1 the pseudo neutrosophic lattice graph of type II.

D1 

 R1

D2   R2

D3   R3

D4   R4

D5 

 R5 D6 

 R6 D7 

D8   R7

D9 

 R8 D10

 R9 D11

H gives the merged neutrosophic graph which can serve as the merged Neutrosophic Relational Maps directed graph of the model. We can use the graph and get the merged connection matrix. 220 Pseudo Lattice Graphs and their Applications to Fuzzy…

It is pertinent to keep on record that we can get any number of experts opinion as the graph and merged connection matrix.

It is pertinent to keep on record that we can get any number of experts opinion as the graph and merged appropriate and get the merged NRMs model.

Such study is time saving and innovative.

Now we give an illustration of how merged graph of two neutrosophic weighted directed graph of the Neutrosophic Relational Equations NREs models can be constructed.

Example 4.18: Let G1 and G2 be two neutrosophic weighted directed graphs associated with the problem given by the experts.

x1  G1 = 0.2  y x2  1 0.3 0.7I x3   y2

0.5  x4  y3 0.6  y4 x5  0.2 0.7 x6 

Pseudo Neutrosophic Lattice Graphs of Type I and II 221

x6  0.7

and G2 = 0.2  y x7  2

0.5

x8   y5 0.3I

x9   y6 0.2

 y7 x10  0.3

Now we can merge only in way to get the pseudo neutrosophic lattice graph of type II. The merged graph is as follows:

x1  0.3 0.2  y x2  1

x3  0.7I  y2 x4  0.7  y3 x5  0.7

0.6  y4 x6  0.2  y5 x7  0.5

x8  0.3I  y6

x9  0.2  y7

0.3 x10  222 Pseudo Lattice Graphs and their Applications to Fuzzy…

Using this graph we can get the merged NREs model. Such study is time saving and annuls any form of discrimination among experts. Interested reader can merge more than two NREs graph and obtain the merged NREs model. Now this is the first time such new study is carried out. An entire chapter is devoted to problems which deal with all types of merging of different models.

Chapter Five

SUGGESTED PROBLEMS

In this chapter we suggest problems some of which are open conjectures and some of them are difficult problems.

1. Let L1 = a1 

a a 3 2    a4

a    a 7 a6 5



a 8 and

b1 

b 2  b3 L2 = be any two lattices.

b   b 5 4

 b6 224 Pseudo Lattice Graphs and their Applications to Fuzzy…

(i) How many single vertex merging pseudo lattice graphs of type I can be obtained? (ii) How many of them in question (i) are lattices?

(iii) How many pseudo lattice graphs of type I can be got merging only two of the vertices and not the edges? (iv) How many of them in question (iii) are lattices? (v) How many pseudo lattice graphs can be got by merging a edge and the two vertices? (vi) How many of them in question (v) are lattices? (vii) How many pseudo lattice graphs of type I can be got by merging only three vertices? (viii) How many pseudo lattice graphs of type I can be got by merging three vertices and two edges? (ix) How many of them are lattices in question (viii).

2. Obtain some special and interesting features enjoyed by pseudo lattice graphs of type I.

a1 3. Let L1 =  a a 3 2    a4

  a7  a a5 6 

a8

and

b1 

b 2  b3 L2 = be any two lattices.

b   b 5 4



b6 Suggested Problems 225

(i) Study questions (i) to (ix) of problem (1) for the pseudo lattice graphs of type I using L1 and L2. (ii) Is the lattice of the pseudo lattice graphs of type I distributive?

(iii) Can we have a pseudo lattice graph of type I lattice to be non modular in this problem?

4. Let L1 =  a1

 a2 a3  a4

 a5

 a6

a a9  a8  7  a10

and  b 1  b2

 b3 L2 =

 b4

 b5

b6   b8

b7   b 9

be two lattices.

226 Pseudo Lattice Graphs and their Applications to Fuzzy…

(a) Study questions (i) to (ix) of problem (1) for this pseudo lattice graph of type one got using lattices L1 and L2. (i) Find all subgraphs of the P the pseudo

lattice graph of type I obtained by merging

vertex a1 with vertex b7. (ii) Can any of these subgraphs of P be lattices? (iii) How many of these subgraphs of P be connected? (iv) Find the total number of subgraphs of P.

b. Study question (i) to (iv) for the B pseudo lattice subgraph of type I where B is obtained by merging one the vertices a1 with b1, a2 with b2, a5 with b3, a6 with b4, a8 with b5 and a10 with b9.

5. Let a1 

 a2

 a  a4 3  a5 L1 = a 6   a7

 a8

  a10  a a9 11

 a12

a13

Suggested Problems 227

and L2 =

b  1

 b2

 b3

 b4 b b 6 5   b7

   b10 b b9 8 

b11 be any two lattices.

(i) In how many ways can L1 and L2 be merged vertices or edges or both to get pseudo lattice graphs of type I? (ii) How many of the pseudo lattice graphs of type I got using L1 and L2 are lattices? (iii) Find all subgraphs of these pseudo lattice graphs of type I which are sublattices. (iv) Is every pseudo lattice graph of type I got using L1 and L2 connected? (v) Does the number pseudo lattice graphs of type I dependent on the number of edges and vertices of the lattices L1 and L2.

b 6. Let L = and L = 1 1  a1 2 

 a2   b2 b3

 a3  b4

be two lattices. 228 Pseudo Lattice Graphs and their Applications to Fuzzy…

Study questions (i) to (v) for this pair of lattices.

7. Let M1 =  a1 and M2 = b1   a2

  b b2 3  a3   a4 b4

be any two lattices.

(i) Study questions (i) to (v) for this pair of lattices. (ii) Compare this pair with the pair in problem (6)

a1 b1 8. Let N1 =  and N2 = 

  a    b a2 3 b2 b3 4

 

a4 b5

be any two lattices.

(i) Study questions (i) to (v) of problem 6 for this pair of lattices. (ii) Compare the pairs in problem 6 and 7 with this pair.

a 1  b0 9. Let L1 = and L2 =   a2 a3  b1

 a4    b2 b3 b4  a5  b5  a6  b6 Suggested Problems 229

be any two lattices.

Study questions (i) to (v) for problem 6 for this pair of lattices L1 and L2.

Compare this pair with M1 and M2 the pair of problem 7.

a1 b1   10. Let L1 = and L2 = a3   a2  b2 b3   a4     a7 a6 a5 b4

 a8 be any two lattices.

(i) Study questions (i) to (v) for problem 6 for this pair. (ii) Does this have any impact on Boolean algebras?

 a1 11. Let L1 = a3 

 a2

a4 

 a5

and b 1

   b2 b3 b4 L2 =  b5 be a pair of lattices.

230 Pseudo Lattice Graphs and their Applications to Fuzzy…

(i) Study questions (i) to (v) of problem 6 for this pair. (ii) Does this have any impact on the non modularity of lattices or modularity of lattices?

12. Obtain some special and interesting features associated with pseudo lattice graphs of type I.

13. Let B1 =  a 1  a2

a4 a 3    a5

a    a 8 a7 6

 a 9

a11  a10

 a12

B2 =  b1

 b2

 b3  c1

 b4 c3 

and B3 =  c2  b5

c4 

 b6  c5

 b7

 b8

Suggested Problems 231

be three lattices.

(i) How many pseudo lattice graphs of type I can be obtained;

(a) by merging only one vertex to each? (b) by merging two vertices or an edge with two vertices each. (c) How many pseudo lattice graphs of type I can be got by merging in all possible ways? (ii) How many of the pseudo lattice graphs of type I using B1, B2 and B3 are lattices?

 a1 b1 14. Let C1 = C2 =  a3      a2 b2 b3 b4

a4    b5 a5

and C3 = c1 

   c  c2 c3 4 c5 

c6

be three lattices.

Study questions (i) to (ii) of problem 13 for these three lattices C1, C2 and C3. b1 

  b 15. Let B1 =  a1 B2 = b2 3

 a2  b4 232 Pseudo Lattice Graphs and their Applications to Fuzzy…

c1 

c3 c2    c4 B3 =    c7 c6 c5

 c8

be three lattices.

(i) Study questions (i) to (ii) of problem 13 for these three lattices.

(ii) Does this happen to have a flavour of Boolean algebras?

b 1

16. Let L1 =  a1 and L2 =   b2 b3  a2  b4  c1 d 1 c3   L3 = L4 =  c 2 d3 d2   d c4  4  c5    d7 d6 d5

 d8

be four lattices.

Suggested Problems 233

Study questions (i) to (iii) of problem 13 for this pair of lattices L1, L2, L3 and L4.

17. Suppose we have n finite lattices L1, L2, …, Ln.

(i) How many pseudo lattice graphs of type I can be constructed? (ii) Study questions (i) to (ii) of problem 13 for these lattices. (iii) How many of these are lattices?

18. Obtain some special features enjoyed by pseudo lattice graphs of type II.

19. Find the major difference between pseudo lattice graphs of type I and type II.

20. Let a1  b1    a L1 = a 2 3 and G2 =



a4 b2 b5  

b b b3   4 b6   7

 b9  b8

be a lattice and a graph respectively.

(i) Let S be the collection of all pseudo lattice graphs of type II. What is the cardinality of S? (ii) How many graph in S are lattices? (iii) Can S have semilattices? (iv) Find all special properties enjoyed by the elements of S. 234 Pseudo Lattice Graphs and their Applications to Fuzzy…

21. Let  a1  v1

L = and G =  a2

a  3 v2   v3

 a4  a5

be a lattice and a graph respectively.

Study questions (i) to (iv) of problem 20 for this G and L.

22. Let v 1  v2 v5  G = 

 v3  v4

be a graph and

a1 

a3 a2    a L = 4

a    a 7 a6 5

 a8

be a lattice.

Study questions (i) to (iv) of problem 20 for this G and L. Suggested Problems 235

23. Let v1  v2 v5 G =

v4

v3 

be a graph and  a1

a 3  L =  a2

a4   a5 be a lattice. Study questions (i) to (iv) of problem 20 for this G and L.

a 1 24. Let G =  v1 and   a v a2 3 6  v2  a4 v a 5  v3 5   a6

 v 4 a9   a7

 a8

be a graph and a lattice respectively.

Study questions (i) to (iv) of problem 20 for this G and L.

236 Pseudo Lattice Graphs and their Applications to Fuzzy…

25. Let v v2 1 

G =

v4 v3

be a graph and  a1

L = a3   a2

a4   be a lattice. a5 S = {Collection of all pseudo lattice graphs of type II}. Study questions (i) to (iv) of problem 20 for this L and G.

w 26. Let 1 v1

v5 w2  w3 G1 =  v2 and G2 = w5  w4

v4 v3 w 6   w7 w8

be two graphs. B = {Collection of all pseudo lattice graph of type II}.

(i) Find o(B). (ii) Find all lattices in B. (iii) Is it possible B contains lattices? (iv) How many subgraph of graphs in B are lattices? (v) Is it possible for B to have semi lattices? Suggested Problems 237

27. Let v1   v2  v3 G1 =

v and 5  v4

w1

 w2  w5  G2 =

 w3  w4 be two graphs. T = {Collection of pseudo lattice graphs of type II got by merging differently G1 with G2}. Study questions (i) to (v) of problem 26 for this T.

v v2  28. Let G1 = 1 

v4  v3  v5

v6  v7 

v10 v11 v8   v9

w1  G2 = w2  w5 

 w3  w4

238 Pseudo Lattice Graphs and their Applications to Fuzzy…

 u1

G = 3 u3 u4    u2

 u5

be three graphs. P = {Collection of all pseudo lattice graphs of type II obtained by merging vertex or edge or both or vertices and edges of the three graphs}

(i) Study questions (i) to (v) of problem 26 for this P. (ii) Compare T of 27 with this P.

29. Let v1  v5 

v4 v6 G1 = v2     v7



v3

w1   u1 G2 = w5 G3 = u2     w2 u4 u5 u3

w4  w 3

be the three graphs. S = {Collection of all pseudo lattice graphs of type II got by merging vertices or edges or both of these graphs}. Study questions (i) to (v) of problem 26 for this S.

Suggested Problems 239

30. Let G1, G2, G3, …, Gn be n graphs. Suppose T = {Collection of all pseudo lattice graphs of type II got by merging vertices or edges or both}.

(i) Study all properties associated with T.

(ii) Study questions (i) to (v) of problem 26 for this T.

31. Let G1 and G2 be two directed graphs associated with FCMs associated a same problem.

 c2 c3  c1  G1 =  c5 c4 c7 

c6 

and

c1   c2

G2 = c8   c9

 c10

We get S is the only pseudo lattice graph of type II obtained by merging c1c2 of G1 with c1c2 of G2. The resultant S gives the FCMs model which is the merged FCMs. Study the merged FCMs in case of any real world problem using 2 or more experts opinion on a same problem.

240 Pseudo Lattice Graphs and their Applications to Fuzzy…

32. Study the merits of using merged FCMs of several FCMs of a problem.

c1 33. Let

c2

G1 = c3 c 4

c1 c5

G2 = c6 c7

c5 c7

G3= c8 c10

c 9

be the three directed graphs of the same problem given by three experts.

Find the merged FCMs of the three directed graphs.

34. If G1, G2, G3, …, Gn be the directed graphs of FCMs related with a problem given by n experts.

Prove there exists more than one merged FCM which can simultaneously give the opinion of the n experts. Suggested Problems 241

35. Let D1 

 R1

D2 

 R 2 D3   R3 G1 = D4   R 4 D5 

 R5 D6 

D7 

and D 1   R1

D8  G2 =  R6

D9   R7

D10  R8

be the directed graphs related to FRMs given by two different experts.

(i) Prove there exists a unique merged graph which gives the merged FRMs.

36. Find / Give some important properties enjoyed by the merged FRMs of n-different experts.

37. Prove such merged FRMs are better than studying n-experts FRMs separately. 242 Pseudo Lattice Graphs and their Applications to Fuzzy…

D1  38. Let  R1

D2  G1 =  R2

D3   R3

D4 

D5 

D2 

 R2

D6  G2 = D7   R4

D8 

 R5

and D8   R4

G3 = D9   R6

D10  R7

be the 3 directed graphs given by three different experts on the same problem using FRMs.

(i) Show there exist one and only one merged FRM of G1, G2 and G3. (ii) Prove this merged FRM is a powerful tool which saves money and economy.

Suggested Problems 243

39. Find some special features enjoyed by merged linked FRMs.

Show merged linked FRMs are different from the merged FRMs and linked FRMs.

D1  40. Let  R 1 D2   R2 G1 = D3   R3

D4   R4

D5 

R1   P1

R2   P2 G2 = R3   P3

R4   P4

R 5 

 P5 R6 

be the two directed graphs of FRM on the same problem given by two different experts.

Find the merged linked FRMs and its directed graph. Show the merged graph is unique. 244 Pseudo Lattice Graphs and their Applications to Fuzzy…

41. Let H1 and H2 be the two bipartite graph given by two experts for the FRM model.

Let G1 

 R1

G2 

 R2 H1 = G3   R 3 G4   R 4 G5 

R1 

 D1

R2  H = 2  D 2 R3   D3 R4   D4

 D 5

 D6

Find the linked graph of G1, G2 got by merging the vertices R1, R2, R3 and R4 of G1 with G2. Show such merging is unique.

42. Obtain some special features enjoyed by merged FRMs.

43. Give by an example 3 or more directed graphs of a FRM can be merged to get a merged FRM. Suggested Problems 245

Show this new merged FRM is a best special type of combined FRM model which save time and money.

44. Study merged Fuzzy Relational Equations (FREs) model.

45. Give some examples of merged FREs of more than two experts.

46. Let x1  0.2

 y1 x 2  0.3 0.5  y2 x3  0.7 G1 =  y3

x4  0.4  y4

x5  1 0.8  y5 x6  0.9

x7 

and

x1  0.2  y1 G2 = 0.5 x9  0.3  y6 0.7 x10  0.9  y7

be the directed graphs associated with the Fuzzy Relational Equations (FRE) model given by two experts who work on the same set of constraints. Prove there exists a unique merged FRE model.

246 Pseudo Lattice Graphs and their Applications to Fuzzy…

47. Study the special features of merged FRE model.

48. Compare a merged FRE model with a linked FRE model.

49. Give one real world problem illustration of merged FRE model.

50. Prove merged FRE models of more than 3 experts by illustrative examples.

51. If G1, G2, …,. Gn are n directed graphs of the FRMs of n experts opinion on a problem.

(i) Prove there exists many merged FRMs. (ii) Prove this merged FRMs is better than using n- FRMs. (iii) Prove this gives equal importance to all the n experts.

52. Obtain some special properties enjoyed by pseudo neutrosophic lattice graphs of type I.

53. Compare the pseudo neutrosophic lattice graphs of type I with that of the pseudo lattice graphs of type II.

54. Enumerate some new and innovative applications of pseudo neutrosophic lattice graphs of type I.

55. Can we merge a real vertex of a lattice with the neutrosophic vertex of another lattice?

56. Can the real edge of a lattice L1 be merged with the neutrosophic edge of the lattice L2?

57. What is the specialty in the case of problems 55 and 56?

Suggested Problems 247

58. Let 1+I 

a1I   a1

L1 = a2I   a2

a3I   a3

 0

and a1 

a2   a3 L = 2  a 4  a5

a6   a7  a8

 a9  a10

be any two neutrosophic lattices.

(i) S = {Collection of all pseudo neutrosophic lattice graphs of type I} find o(S). (ii) How many of these A  S are lattices? (iii) Is every A  S a connected graph?

59. Let a1 

a2 a4 a5 a6 L1 =  a3     a7 248 Pseudo Lattice Graphs and their Applications to Fuzzy…

and b1 

b4 b2   b3 

L2 = b7   b6  b5  b8

be any two neutrosophic lattices.

(i) P = {Collection of all pseudo neutrosophic lattice graphs of type I}; find o(P). (ii) Find all neutrosophic lattices in P.

60. Obtain some special features enjoyed by pseudo neutrosophic lattice graphs of type II.

61. Is it possible that by merging vertices or edges or both of two neutrosophic lattices? The resultant pseudo neutrosophic lattice graph of type II has no lattice and only graphs?

62. Let a1 

a a a5 a6 L1 = 2  a3  4    a5

and v1  v2  v5  G =

 v3  v4 be a neutrosophic lattice and a neutrosophic graph respectively.

Suggested Problems 249

(i) Find order of S = {Collection of a pseudo neutrosophic lattice graphs of type II}. (ii) How many elements in S are neutrosophic lattices?

63. Let L and L be any two finite neutrosophic lattices. 1 2 S = {Collection of all edge or vertex or merging of both of the lattices L1 and L2}. = {Collection of all pseudo neutrosophic lattice graphs of type I}.

(i) Find o(S). (ii) Does order of S depend on the number of vertices and edges of L1 and L2? (iii) Obtain some special features enjoyed by S. (iv) Prove S can have subgraphs / sublattices which are not neutrosophic.

64. Let a1 

a2   a3  a4 L1 =  a5

 a6

 a7

and  b1

 b2

 b3 L2 =  b4

b5   b6

 b7

250 Pseudo Lattice Graphs and their Applications to Fuzzy…

be neutrosophic lattices. S = {Collection of all pseudo neutrosophic lattice graphs of type I got by merging the vertices or edges or both}.

(i) Find o(S). (ii) Find all lattices in S. (iii) Find sublattices A in S.

65. Let a1 

a a 3 2    a4 L1 =

a    a 7 a6 5

 a8 and  b1

b3 

L2 =  b2

b4 

 b5

be any two lattices. S = {Collection of all pseudo neutrosophic lattice graphs of merging vertices or edges of L1 with L2}.

(i) Find o(S). (ii) Prove S has sublattices. (iii) Can A  S be a lattice?

66. Let a1  L = a2   a3  a4

 a5 Suggested Problems 251

and  v1

G =

v2   v3

be the lattice and the neutrosophic graph. S = {Collection of all pseudo neutrosophic lattice graphs of type II}.

Study questions (i) to (iii) of problem 65 for this S.

67. Let a1  a a2   a3  a4  a5  6 L =  a7 a9   a8

 a10

and v 1  v 2 v5  G = 

 v3  v4

be a neutrosophic lattice and graph. Study questions (i) to (iii) of problem 65 for this S.

68. Let L and G be a neutrosophic lattice and a graph respectively given in the following.

252 Pseudo Lattice Graphs and their Applications to Fuzzy…

 v1 a1  v a a 2 2   a3  4  v5 

 a 5  a  v3 6  v4  a7

Study questions (i) to (iii) of problem (59) for this P = {Collection of all pseudo lattice graphs of type II using L and G.

69. Let G and G1 be two graphs where G1 is the usual graph.

v1  v2 v5  G1 = 

 v3  v 4

and v1 v5 v  8   v4 v7 G2 = v2   

 v9   v6 v3

M = {Collection of all pseudo lattice graphs of type II using G1 and G2}. Study question (i) to (iii) of problem 65 for this M.

Suggested Problems 253

70. Let v1 

G1 = v2 v3  

v6   v5 v4   v9

  v   11 v10 v v8 7

v1 and 

u3   u2 G2 = u u8   u4  5

u7   u6

 u9

be two neutrosophic graphs. S = {Collection of all pseudo lattice graphs of type II}.

Study question (i) to (iii) of problem 65 for this S.

71. Show pseudo neutrosophic lattice graphs of II are used in neutrosophic fuzzy models like NCMs (Neutrosophic Relational Maps) and NREs (Neutrosophic Relational Equations).

72. Give one example from real word problem where the use of NCMs directed graphs are merged to get the merged neutrosophic cognitive Maps model.

254 Pseudo Lattice Graphs and their Applications to Fuzzy…

73. Explain by illustration the merged NRMs model.

74. Describe by an example the merged NRE model.

75. Let G1 =

C 1

C4

C2

C3

and G2 =

C2 C1

C8

C6

C7

be the direct graphs of the FCM and NCM respectively given by two experts on the same problem.

We can merge G1 and G2 only in one way by merging the edge C1C3. The resultant gives the merged NCM.

Study the using the merged graphs connection matrix the merged NCM model. Suggested Problems 255

76. Let G1 and G2 be two directed neutrosophic graphs associated with the NCMs model for the same problem.

Prove the merged directed graph is unique and can be got

only by merging C1 node of G1 with C1 node of G2.

Study the merged NCMs model.

77. Let

D1 

 R1

D2   R2 G1 = D3   R3 D4   R4

D5 

and

D5   R3

G2 = D6   R5

 D7  R6

 D R7 8 

be the neutrosophic directed bipartite graphs of the NRMs model given by two experts on the same problem.

(i) Show we have only one merged NRM. (ii) We can have one and only pseudo lattice graph of type II. (iii) Study the merged NRM model. 256 Pseudo Lattice Graphs and their Applications to Fuzzy…

78. Let D1 

 R1

D2  G1 =  R2

D3   R3

D4   R4

D5 

D1   R2

D6 

G2 =  R5

D7 

 R6

D8 

and

D8   R2

G3 = D9   R6

D10   R7

 D11  R8

 R9 D12 

be any three directed neutrosophic bipartite graphs of a NRM model which is given above.

Suggested Problems 257

(i) Show the merged NRM is unique. (ii) Study the merged NRM model. (iii) What are the merits of merged NRMs model? (iv) Show it is different from combined NRMs model and linked NRMs model.

79. Let x1  0.3

0.1  y x2  1 0.7 0.2 G1 = x3   y2 0.1 0.8 x  y3 4  0.6

 y4 x5  0.7

and x5  0.7

0.8  y x6  4 0.4 0.4 G = 2  y x7  5 0.9I 0.5 x  y6 8  0.2I

0.7 x9 

be the bipartite graph of the FRE and NRE respectively given in the following.

(i) Show there exist one and only one merged NRE using G1 and G2. (ii) Find some special features about these merged NREs. 258 Pseudo Lattice Graphs and their Applications to Fuzzy…

(iii) What are the basic advantages in using the merged NRE models.

74. Let

x1  0.3 0.2  y x2  1 0.9 0.3I x  y2 G1 = 3  0.8

0.9  y x4  3 0.8 0.3  y4 x5  0.9I

0.2 x6 

and x1  0.8I  y1 x2   y3 G = 0.3I 2 x3   y5 0.7I x4   0.3 y6

x5  0.9  y7

0.6  y 8

be the neutrosophic bipartite graphs of two NREs which are related with the same problem.

(i) Prove there exists only one merged neutrosophic graph describing the merged NRE model. Suggested Problems 259

(ii) Study this NRE model (iii) Spell out the advantages of using the merged NREs model.

81. Give a real world problem in which merged NREs model is used.

82. Enumerate the general merits of using merged fuzzy or neutrosophic models.

83. Differentiate the merged FCMs models and the combined FCMs models.

84. What is the difference between the merged FRMs model and linked FRMs model?

Study the above question in NRM models.

85. Distinguish the properties between the merged NRMs model and the combined NRM model.

86. Give some interesting applications of pseudo lattice graphs of type II.

87. Show by the method of merging models one can merge more than two graphs to get the merged model if the necessary conditions are satisfied.

88. Give a real world problem in which the three graphs of a FCMs model are merged on in the concepts and not on the edges. Study the same problem in case of NCMs.

89. Can problem 85 be true in case of related graphs of FRMs and NRMs?

90. Give a real world problem illustrated in which four appropriate graphs of NREs / FREs are merged to get the new merged NRE model. 260 Pseudo Lattice Graphs and their Applications to Fuzzy…

(i) Study the advantages. (ii) What can be the probable disadvantages in using merged fuzzy models?

91. Can the pseudo lattice graphs of type I or type II which are trees be helpful in data mining?

92. Obtain a sufficient and necessary condition for the pseudo lattice graph to be a lattice?

93. Is it possible if two lattices are used then no pseudo lattice graph of type I will be a lattice?

94. Suppose we use a merged FCM (or FRM or FRE) model using the merging of their respective graph.

(i) Can we say certain special subgraphs of the merged graphs result in submodels?

(ii) Can we always say subgraphs of every experts can be got from the merged model?

(iii) Prove use of merged models save time and economy.

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75. Vasantha Kandasamy and Smarandache Florentin, Analysis of social aspects of migrant labourers living with HIV/AIDS using Fuzzy Theory and Neutrosophic Cognitive Maps, Xiquan, Phoenix, 2004.

76. Vasantha Kandasamy, W.B. and Florentin Smarandache, Introduction to n-adaptive Fuzzy Models to Analyze Public opinion in AIDS, Hexis, Arizona, 2006.

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86. Vasantha Kandasamy, W.B., and Praseetha, R., New Fuzzy Relation Equations to Estimate the Peak Hours of the Day for Transport Systems, J. of Bihar Math. Soc., 20 (2000) 1- 14.

87. Vasantha Kandasamy, W.B., Vasuki, R., and Thulukkanam, K., Kosko Hamming Distance in the analysis of FCMs to study the problems of locals due to dumping of solid waste in Kodungaiyur, Ultra Scientist, 26 (2014) 55-62.

88. Vasantha Kandasamy, W.B., and Uma, S., Combined Fuzzy Cognitive Map of Socio-Economic Model, Appl. Sci. Periodical, 2 (2000) 25-27.

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93. Vasantha Kandasamy, W.B., and Yasmin Sultana, FRM to Analyse the Employee-Employer Relationship Model, J. Bihar Math. Soc., 21 (2001) 25-34.

94. Vasantha Kandasamy, W.B., and Yasmin Sultana, Knowledge Processing Using Fuzzy Relational Maps, Ultra Sci., 12 (2000) 242-245.

95. Vasantha Kandasamy, W.B., Florentin Smarandache and K. Ilanthenral, Elementary Fuzzy Matrix Theory and Fuzzy Models for Socio Scientists, Automaton, 2007.

96. Vasantha Kandasamy, W.B., Mary John, M., and T. Kanagamuthu. Study of Social Interaction and Woman Empowerment Relative to HIV/AIDS, Maths Tiger, 1(4) (2002) 4-7.

97. Vasantha Kandasamy, W.B., Neelakantan, N.R., and S. Ramathilagam. Maximize the Production of Cement Industries by the Maximum Satisfaction of Employees using Fuzzy Matrix, Ultra Science, 15 (2003) 45-56.

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103. Zadeh, L.A., A Theory of Approximate Reasoning, Machine Intelligence, 9 (1979) 149- 194.

104. Zimmermann, H.J., Fuzzy Set Theory and its Applications, Kluwer, Boston, 1988.

INDEX

C

Chain lattice, 9 Connected pseudo lattice subgraphs of type II, 131-5

D

Disconnected subgraphs of pseudo lattice graphs of type II, 132- 7 Distributive lattice, 9-10

G

Graph, 7

L

Lattice, 7

M

Merged linked NRMs, 215-7 Merged NCMs, 209-12 Merged NREs model, 219-222 Merged or glued FCMs, 162-5 Merged or glued FRMs, 165-9

Index 275

N

Neutrosophic Cognitive Maps (NCMs) model, 201-9 Neutrosophic connection matrix, 201-9 Neutrosophic graphs, 179-185 Neutrosophic lattices, 179-186 Neutrosophic Relational Equations, 219 221 Neutrosophic subgraphs, 189-192 NRMs, 215-7

P

Peterson graph, 113-4 Pseudo neutrosophic lattice graphs of type II, 179-188 Pseudo lattice graph of type I, 8-70 Pseudo lattice graph of type II, 99-120 Pseudo lattice subgraph of type I, 82-95 Pseudo lattice subgraph of type II, 129-37 Pseudo neutrosophic lattice graphs of type I, 179-185 Pseudo neutrosophic graphs of type II, 179-185 Pseudo neutrosophic lattice subgraphs of type I, 186-8

S

Strong merged pseudo lattice graphs of type II, 154-6

U

Unnatural or cranky neutrosophic pseudo lattice graphs, 192-5

ABOUT THE AUTHORS

Dr.W.B.Vasantha Kandasamy is a Professor in the Department of Mathematics, Indian Institute of Technology Madras, Chennai. In the past decade she has guided 13 Ph.D. scholars in the different fields of non-associative algebras, algebraic coding theory, transportation theory, fuzzy groups, and applications of fuzzy theory of the problems faced in chemical industries and cement industries. She has to her credit 653 research papers. She has guided over 100 M.Sc. and M.Tech. projects. She has worked in collaboration projects with the Indian Space Research Organization and with the Tamil Nadu State AIDS Control Society. She is presently working on a research project funded by the Board of Research in Nuclear Sciences, Government of India. This is her 97th book. On India's 60th Independence Day, Dr.Vasantha was conferred the Kalpana Chawla Award for Courage and Daring Enterprise by the State Government of Tamil Nadu in recognition of her sustained fight for social justice in the Indian Institute of Technology (IIT) Madras and for her contribution to mathematics. The award, instituted in the memory of Indian-American astronaut Kalpana Chawla who died aboard Space Shuttle Columbia, carried a cash prize of five lakh rupees (the highest prize-money for any Indian award) and a gold medal. She can be contacted at [email protected] Web Site: http://mat.iitm.ac.in/home/wbv/public_html/ or http://www.vasantha.in

Dr. Florentin Smarandache is a Professor of Mathematics at the University of New Mexico in USA. He published over 75 books and 200 articles and notes in mathematics, physics, philosophy, psychology, rebus, literature. In mathematics his research is in number theory, non-Euclidean geometry, synthetic geometry, algebraic structures, statistics, neutrosophic logic and set (generalizations of fuzzy logic and set respectively), neutrosophic probability (generalization of classical and imprecise probability). Also, small contributions to nuclear and particle physics, information fusion, neutrosophy (a generalization of dialectics), law of sensations and stimuli, etc. He got the 2010 Telesio-Galilei Academy of Science Gold Medal, Adjunct Professor (equivalent to Doctor Honoris Causa) of Beijing Jiaotong University in 2011, and 2011 Romanian Academy Award for Technical Science (the highest in the country). Dr. W. B. Vasantha Kandasamy and Dr. Florentin Smarandache got the 2012 New Mexico-Arizona and 2011 New Mexico Book Award for Algebraic Structures. He can be contacted at [email protected]

K. Ilanthenral is the editor of The Maths Tiger, Quarterly Journal of Maths. She can be contacted at [email protected]