FUZZY IDEALS OF SOME ALGEBRAS
AREEJ TAWFEEQ HAMEED AL-BUDARUB;
Abstract
The notion of BCK-algebras introduced the class of abstract algebras BCK-algebras as a generalization of the concept of set-theoretic difference and propositional calculus . In 1966, (Y. Iami and K. Iseki) introduced the notion of BCK-algebra, and the notion of BCI-algebra which is a generalization of BCK-algebra ([23], [25]). Since the numerous mathematical papers have been written investigating the algebraic properties of the BCK / BCI-algebras , there is a great deal of literature has been produced on the theory of BCK/BCI-algebras, emphasis seems to have been put on the ideal theory of BCK/ BCI-algebras ([23], [25],[31]). In 1983, (Q. Hu and X. Li) ([21],[22]) introduced the notion of a BCH-algebra which is a generalization of the notion of BCI-algebras and studied a few properties of these algebras . In 1983 , (Y. Komori) [46] introduced a notion of BCC-algebras, and (W. A. Dudek) ([16],[17]) redefined the notion of BCC-algebras by using a dual form of the ordinary definition in the sense of Y. Komori. In [13] ,(W. A. Dudek and X. H. Zhang) introduced a notion of BCC-ideals in BCC algebras and described connections between such ideals and congruences. The concept of fuzzy sets was first introduced by (L.A. Zadeh) [78],in 1965 . From that time, the theory of fuzzy sets has been developed in many directions and found applications in a wide variety of fields . It application to various mathematical contexts has given rise to what is now commonly called fuzzy mathematics. Fuzzy algebras is an important branch of fuzzy mathematics, they have studied some algebraic structures such as fuzzy groups by (A. Rosenfeld ) [69] in 1971 . In the time , , they have studied some algebraic structures such as fuzzy rings, fuzzy ideals, fuzzy vectors spaces. In 1975 , (L.A. Zadeh) [79] introduced the notion of an interval-valued fuzzy BCK-ideal (briefly i-v fuzzy BCK-ideal) . In 1991, (O. Xi) [74] applied the concept of fuzzy BCK-algebras . (J. Meng and Y.B. Jun) [52] in 1994, studied the fuzzification of BCK-algebra and BCI-algebra. In 2001 (Y. B. Jun) [13] introduced fuzzy BCC-subalgebras and investigated some related properties.
In 2001, (J.Neggers, S.S.Ahn and H.S.Kim) [64] introduced a new notion, called a Q-algebra and generalized some theorems discussed in BCI/BCK-algebras. In 2002, (J. Neggers and H. S. Kim) [61] introduced a new notion, called a β-algebra and obtained several results. In 2009 , (S.M. Mostafa , M.A. Abd-Elanaby and O.R.M. Elgendy) [58] introduced a KU-ideals of KU-algebra . In 2009 , (C. Prabpayak and U. Leerawat) [66] introduced a new algebraic structure which is called KU-algebras , and studied ideals and congruences in KU-algebras . They gave the concept of homomorphism of KU-algebras and investigated some related properties ([66],[67]).
Summary of the Thesis:
The main objective of this thesis is to study new structures of two algebras called (KUS-algebras and SA-algebras). Furthermore a new definitions of ideals of its , fuzzy ideals of its are investigated, examples , propositions and several theorems are stated and proved.
The whole work is divided into four chapters.
Chapter 1 :
This chapter is devoted to the basic definitions and results concerning several types of algebras and its Fuzzy sets ,which are required in the succeeding sections. All results here are quoted from existing literature.
Chapter 2 :
This chapter consists of two sections. In the first section, we introduce a new notion of algebras called KUS-algebras. The notions of KUS-algebras, KUS-subalgebras , KUS-ideals and homomorphism of KUS-algebras are introduced. The relation between some abelian groups and KUS-algebras are studied and investigated some of its properties .
In the second section, we introduce the notion of fuzzy KUS-ideal of KUS-algebra, several theorems, properties are stated and proved.
Chapter 3 :
This chapter consists of two sections. In the first section, the notion of interval-valued fuzzy KUS-ideals (briefly i-v fuzzy KUS-ideal) on KUS-algebras is introduced. Several theorems of i-v fuzzy KUS-ideal are stated and proved. The image and inverse image of i-v fuzzy KUS-ideals under homomorphism are defined and how the homomorphic images and inverse images of i-v fuzzy KUS-ideals become i-v fuzzy KUS-ideals of KUS-algebras is studied as well.
In the second section, we introduce the notion of anti-fuzzy KUS-ideals of KUS-algebras and then we study the homomorphism image and inverse image of anti-fuzzy KUS-ideals. We also prove that the Cartesian product of anti-fuzzy KUS-ideals is an anti-fuzzy KUS-ideals .
Chapter 4 :
This chapter consists of two sections. In the first section, we introduce a new algebraic structure with two operations, which is called SA-algebras, we define the SA-ideals of SA-algebras and investigate some of its properties. Furthermore, we described the relation between ideals and congruence’s and we consider some relations between these ideals and quotient algebras that are obstructed via these ideals.
In the second section, we introduce the notion of fuzzy SA-ideal with degree (λ,κ) of SA-algebra, several theorems and properties are stated and proved . we study the fuzzy relations on fuzzy SA-ideal with degree (λ,κ) of SA-algebras are also studied.
Remark:
The results of chapters ( 2, 3, 4) are published in international Journals (see the end the of Thesis page - 140- ) .
In 2001, (J.Neggers, S.S.Ahn and H.S.Kim) [64] introduced a new notion, called a Q-algebra and generalized some theorems discussed in BCI/BCK-algebras. In 2002, (J. Neggers and H. S. Kim) [61] introduced a new notion, called a β-algebra and obtained several results. In 2009 , (S.M. Mostafa , M.A. Abd-Elanaby and O.R.M. Elgendy) [58] introduced a KU-ideals of KU-algebra . In 2009 , (C. Prabpayak and U. Leerawat) [66] introduced a new algebraic structure which is called KU-algebras , and studied ideals and congruences in KU-algebras . They gave the concept of homomorphism of KU-algebras and investigated some related properties ([66],[67]).
Summary of the Thesis:
The main objective of this thesis is to study new structures of two algebras called (KUS-algebras and SA-algebras). Furthermore a new definitions of ideals of its , fuzzy ideals of its are investigated, examples , propositions and several theorems are stated and proved.
The whole work is divided into four chapters.
Chapter 1 :
This chapter is devoted to the basic definitions and results concerning several types of algebras and its Fuzzy sets ,which are required in the succeeding sections. All results here are quoted from existing literature.
Chapter 2 :
This chapter consists of two sections. In the first section, we introduce a new notion of algebras called KUS-algebras. The notions of KUS-algebras, KUS-subalgebras , KUS-ideals and homomorphism of KUS-algebras are introduced. The relation between some abelian groups and KUS-algebras are studied and investigated some of its properties .
In the second section, we introduce the notion of fuzzy KUS-ideal of KUS-algebra, several theorems, properties are stated and proved.
Chapter 3 :
This chapter consists of two sections. In the first section, the notion of interval-valued fuzzy KUS-ideals (briefly i-v fuzzy KUS-ideal) on KUS-algebras is introduced. Several theorems of i-v fuzzy KUS-ideal are stated and proved. The image and inverse image of i-v fuzzy KUS-ideals under homomorphism are defined and how the homomorphic images and inverse images of i-v fuzzy KUS-ideals become i-v fuzzy KUS-ideals of KUS-algebras is studied as well.
In the second section, we introduce the notion of anti-fuzzy KUS-ideals of KUS-algebras and then we study the homomorphism image and inverse image of anti-fuzzy KUS-ideals. We also prove that the Cartesian product of anti-fuzzy KUS-ideals is an anti-fuzzy KUS-ideals .
Chapter 4 :
This chapter consists of two sections. In the first section, we introduce a new algebraic structure with two operations, which is called SA-algebras, we define the SA-ideals of SA-algebras and investigate some of its properties. Furthermore, we described the relation between ideals and congruence’s and we consider some relations between these ideals and quotient algebras that are obstructed via these ideals.
In the second section, we introduce the notion of fuzzy SA-ideal with degree (λ,κ) of SA-algebra, several theorems and properties are stated and proved . we study the fuzzy relations on fuzzy SA-ideal with degree (λ,κ) of SA-algebras are also studied.
Remark:
The results of chapters ( 2, 3, 4) are published in international Journals (see the end the of Thesis page - 140- ) .
Other data
| Title | FUZZY IDEALS OF SOME ALGEBRAS | Other Titles | المثاليات الفازية في بعض الجبريات | Authors | AREEJ TAWFEEQ HAMEED AL-BUDARUB | Issue Date | 2015 |
Recommend this item
Similar Items from Core Recommender Database
Items in Ain Shams Scholar are protected by copyright, with all rights reserved, unless otherwise indicated.