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Interval Semigroups INTERVAL SEMIGROUPS W. B. Vasantha Kandasamy Florentin Smarandache 2011 INTERVAL SEMIGROUPS W. B. Vasantha Kandasamy Florentin Smarandache 2011 2 CONTENTS Preface 5 Dedication 6 Chapter One INTRODUCTION 7 Chapter Two INTERVAL SEMIGROUPS 9 Chapter Three INTERVAL POLYNOMIAL SEMIGROUPS 37 Chapter Four SPECIAL INTERVAL SYMMETRIC SEMIGROUPS 47 Chapter Five NEUTROSOPHIC INTERVAL SEMIGROUPS 61 3 Chapter Six NEUTROSOPHIC INTERVAL MATRIX SEMIGROUPS AND FUZZY INTERVAL SEMIGROUPS 73 6.1 Pure Neutrosophic Interval Matrix Semigroups 73 6.2 Neutrosophic Interval Polynomial Semigroups 94 6.3 Fuzzy Interval Semigroups 118 Chapter Seven APPLICATION OF INTERVAL SEMIGROUPS 129 Chapter Eight SUGGESTED PROBLEMS 131 FURTHER READING 159 INDEX 161 ABOUT THE AUTHORS 165 4 PREFACE In this book we introduce the notion of interval semigroups using intervals of the form [0, a], a is real. Several types of interval semigroups like fuzzy interval semigroups, interval symmetric semigroups, special symmetric interval semigroups, interval matrix semigroups and interval polynomial semigroups are defined and discussed. This book has eight chapters. The main feature of this book is that we suggest 241 problems in the eighth chapter. In this book the authors have defined 29 new concepts and illustrates them with 231 examples. Certainly this will find several applications. The authors deeply acknowledge Dr. Kandasamy for the proof reading and Meena and Kama for the formatting and designing of the book. W.B.VASANTHA KANDASAMY FLORENTIN SMARANDACHE 5 ~ DEDICATED TO ~ Ayyankali Ayyankali (1863–1941) was the first leader of Dalits from Kerala. He initiated several reforms to emancipate the lives of the Dalits. Ayyankali organized Dalits and fought against the discriminations done to Dalits and through his efforts he got the right to education, right to walk on the public roads and dalit women were allowed to cover their nakedness in public. He spearheaded movements against casteism. 6 Chapter One INTRODUCTION We in this book make use of special type of intervals to build interval semigroups, interval row matrix semigroups, interval column matrix semigroups and interval matrix semigroups. We also introduce and study the Smarandache analogue of them. The new notion of interval symmetric semigroups and special interval symmetric semigroups are defined and studied. For more about symmetric semigroups and their Smarandache analogue concepts please refer [9]. The classical theorems for finite groups like Lagrange theorem, Cauchy theorem and Sylow theorem are introduced in a special way and analyzed. Only under special conditions we see the notion of these classical theorems for finite groups can be extended interval semigroups. The authors also introduce the notion of neutrosophic intervals and fuzzy intervals and study them in the context of interval semigroups. I denotes the indeterminate or inderminancy where I2 = I and I + I = 2I, I + I + I = 3I and so 7 on. For more about neutrosophy, neutrosophic intervals please refer [1, 3, 6-8]. Study of special elements like interval zerodivisors, interval idempotents, interval units, interval nilpotents are studied and their Smarandache analogue introduced [9]. 8 Chapter Two INTERVAL SEMIGROUPS In this chapter we for the first time introduce the notion of interval semigroups and describe a few of their properties associated with them. We see in general several of the classical theorems are not true in general case of semigroups. First we proceed on to give some notations essential to develop these new structures. I (Zn) = {[0, am] | am ∈ Zn}, I(Z+ ∪ {0}) = {[0, a] | a ∈ Z+ ∪ {0}}, I(Q+ ∪ {0}) = {[0, a] | a ∈ Q+ ∪ {0}}, I(R+ ∪ {0}) = {[0, a] | a ∈ R+ ∪ {0}} and I(C+ ∪ {0}) = {[0, a] | a ∈ C+ ∪ {0}}. DEFINITION 2.1: Let S = {[0, ai] | ai ∈ Zn; +} S is a semigroup under addition modulo n. S is defined as the interval semigroup under addition modulo n. We will first illustrate this by some simple examples. 9 Example 2.1: Let S = {[0, ai] | ai ∈ Z6}, under additions is an interval semigroup. We see S is of finite order and order of S is six. Example 2.2: Let S = {[0, ai] | ai ∈ Z12} be an interval semigroup under addition modulo 12. This is also a interval semigroup of finite order. Now we can define interval semigroup under addition using Z+ ∪ {0}, Q+ ∪ {0}, R+ ∪ {0} and C+ ∪ {0}. All these interval semigroups are of infinite order. We will illustrate these situations by some examples. + Example 2.3: Let S = {[0, ai] | ai ∈ Z ∪ {0}}; S is an interval semigroup under addition. Clearly S is of infinite order. + Example 2.4: Let S = {[0, ai] | ai ∈ Q ∪ {0}}; S is an interval semigroup under addition. Clearly S is of infinite order. + Example 2.5: Let S = {[0, ai] | ai ∈ R ∪ {0}}; S is an interval semigroup under addition and is of infinite order. + Example 2.6: Let S = {[0, ai] | ai ∈ C ∪ {0}}; S is an interval semigroup under addition and is of infinite order. Thus we have seen examples of interval semigroups under addition, these are known as basic interval semigroups under addition. We will now define polynomial interval semigroups and matrix interval semigroups defined using basic interval semigroups and then define their substructures. DEFINITION 2.2: Let S = {([0, a1], [0, a2], …, [0, an]) | ai ∈ Zn}; S under component wise addition is an interval semigroup known as the row matrix interval semigroup. + + We can in the definition replace Zn by Z ∪ {0} or R ∪ {0} or Q+ ∪ {0} or C+ ∪ {0}. 10 We will illustrate these by some examples. Example 2.7: Let S = {([0, a1], [0, a2], [0, a3], [0, a4], [0, a5]) | ai ∈ Z12; 1 ≤ i ≤ 5} is a row matrix interval semigroup under addition. Example 2.8: Let P = {([0, a1], [0, a2], [0, a3], [0, a4], [0, a5], + [0, a6]) | ai ∈ Z ∪ {0}; 1 ≤ i ≤ 6} is a row matrix interval semigroup under addition. Clearly P is of infinite order. Example 2.9: Let S = {([0, a1], [0, a2], [0, a3], …, [0, a12]) / ai ∈ Q+ ∪ {0}; 1 ≤ i ≤ 12}; S is a row matrix interval semigroup under addition and is of infinite order. Example 2.10: Let S = {([0, a1], [0, a2], [0, a3], …, [0, a15]) / ai ∈ R+ ∪ {0}; 1 ≤ i ≤ 15} be a row matrix interval semigroup under addition; T is of infinite order. + Example 2.11: Let G = {([0, a1], [0, a2]) | ai ∈ C ∪ {0}; 1 ≤ i ≤ 2}; be a row matrix interval semigroup of infinite order. Now we proceed onto define column matrix interval semigroup. DEFINITION 2.3: Let ⎧⎡⎤[0,a ] ⎫ ⎪ 1 ⎪ ⎢⎥[0,a ] ⎪⎢⎥2 ⎪ ⎪ aZ;im∈ ⎪ S = ⎨⎢⎥[0,a3 ] ⎬ , ⎢⎥1in≤≤ ⎪ # ⎪ ⎪⎢⎥ ⎪ ⎢⎥[0,a ] ⎩⎭⎪⎣⎦n ⎪ S under addition modulo m is a semigroup defined as the column interval matrix semigroup under addition. + + + We can replace Zn is definition 2.3 by Z ∪ {0} or Q ∪ {0}, R ∪ {0} or C+ ∪ {0} and get column interval matrix semigroups under addition. 11 We will illustrate these situations by some examples. Example 2.12: Let ⎧⎡⎤[0,a1 ] ⎫ ⎪⎢⎥ ⎪ ⎪ [0,a2 ] ⎪ ⎪⎢⎥ ⎪ S = ⎨⎢⎥[0,a3i5 ] a∈ Z ;1≤≤ i 5⎬ ⎢⎥ ⎪ [0,a ] ⎪ ⎪⎢⎥4 ⎪ ⎢⎥[0,a ] ⎩⎭⎪⎣⎦5 ⎪ be a column interval matrix semigroup under addition. S is of finite order. Example 2.13: Let ⎧⎫⎡⎤[0,a ] ⎪⎪1 ⎢⎥[0,a ] ⎪⎪⎢⎥2 ⎪⎪+ S = ⎨⎬⎢⎥[0,a3i ] a∈∪ Z {0};1 ≤≤ i 15 ⎢⎥ ⎪⎪# ⎪⎪⎢⎥ ⎢⎥[0,a ] ⎩⎭⎪⎪⎣⎦15 be a column interval matrix semigroup under addition. Example 2.14: Let ⎧⎫⎡⎤[0,a ] ⎪⎪1 ⎢⎥[0,a ] ⎪⎪⎢⎥2 ⎪⎪+ S = ⎨⎬⎢⎥[0,a3i ] a∈∪ Q {0};1 ≤≤ i 10 , ⎢⎥ ⎪⎪# ⎪⎪⎢⎥ ⎢⎥[0,a ] ⎩⎭⎪⎪⎣⎦10 be a column interval semigroup under addition of infinite order. 12 Example 2.15: Let ⎧⎡⎤[0,a ] ⎫ ⎪ 1 ⎪ ⎢⎥[0,a ] ⎪⎢⎥2 ⎪ ⎪ + ⎪ S = ⎨⎢⎥[0,a3i ] a∈ R∪≤≤ {0};1 i 6⎬ ⎢⎥ ⎪ # ⎪ ⎪⎢⎥ ⎪ ⎢⎥[0,a ] ⎩⎭⎪⎣⎦6 ⎪ be a column interval semigroup under addition of infinite order. Example 2.16: Let ⎧⎡⎤[0,a ] ⎫ ⎪ 1 ⎪ ⎢⎥[0,a ] ⎪⎢⎥2 + ⎪ P = ⎨ aCi ∈ ∪≤≤ {0};1i4⎬ ⎢⎥ ⎪ [0,a3 ] ⎪ ⎢⎥ ⎪ [0,a ] ⎪ ⎩⎭⎣⎦4 be a column interval semigroup under addition of infinite order. Now we will define matrix interval semigroup. DEFINITION 2.4: Let S = {m × n interval matrices with entries from I(Zn)} be a m × n matrix interval semigroup under addition. + + We can replace I (Zn) in definition 2.4 by I (Z ∪ {0}) or I (R ∪ {0}) or I(Q+ ∪ {0}) or I(C+ ∪ {0}) and get m × n interval matrix semigroups under addition. We will illustrate all these by some examples. Example 2.17: Let ⎧⎫⎡⎤[0,a ] [0,a ] ⎪⎪12 S = ⎢⎥[0,a ] [0,a ] a∈≤≤ Z ;1 i 6 ⎨⎬⎢⎥34i10 ⎪⎪⎢⎥[0,a ] [0,a ] ⎩⎭⎣⎦56 13 be a 3 × 2 interval matrix semigroup under matrix addition modulo 10 of finite order. Example 2.18: Let ⎧⎫⎡⎤[0,a ] [0,a ] [0,a ] ⎪⎪123 S = ⎢⎥[0,a ] [0,a ] [0,a ] a∈ Z ;1≤≤ i 9 ⎨⎬⎢⎥456i42 ⎪⎪⎢⎥[0,a ] [0,a ] [0,a ] ⎩⎭⎣⎦789 be a 3 × 3 interval square matrix semigroup of finite order under interval matrix addition modulo 42. Example 2.19: Let ⎧ ⎡⎤[0,a ] [0,a ] [0,a ] ⎫ ⎪ 123 ⎪ ⎢⎥[0,a ] [0,a ] [0,a ] ⎪ ⎢⎥456 + ⎪ S = ⎨ aQi ∈∪ {0};1i24 ≤≤⎬ ⎪ ⎢⎥### ⎪ ⎢⎥ ⎪ [0,a ] [0,a ] [0,a ] ⎪ ⎩⎭⎣⎦22 23 24 be a 8 × 3 matrix interval semigroup under addition of infinite order. Example 2.20: Let ⎧⎫⎡⎤[0,a ]" [0,a ] ⎪⎪14 ⎢⎥[0,a ]" [0,a ] ⎪⎪⎢⎥58+ P = ⎨⎬aRi ∈∪ {0};1i16 ≤≤ ⎢⎥ ⎪⎪[0,a912 ]" [0,a ] ⎢⎥ ⎪⎪[0,a ]" [0,a ] ⎩⎭⎣⎦13 16 be a 4 × 4 square matrix interval semigroup of infinite order. 14 DEFINITION 2.5: Let S be the matrix interval semigroup under addition. Let M ⊆ S (M a proper subset of S), if M itself is a matrix interval semigroup under addition then we define M to be a matrix interval subsemigroup of S. We will illustrate this situation by some examples. Example 2.21: Let ⎪⎪⎧⎫⎡⎤[0,a12 ] [0,a ] S = ⎨⎬aZ;1i4i5∈≤≤ ⎢⎥[0,a ] [0,a ] ⎩⎭⎪⎪⎣⎦34 be a square matrix interval semigroup under addition.
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