الفهرس 
Sub implicative ideals of KU-AlgebrasOnly 14 pages are availabe for public view
 |  | from | 125 |  |  |
| | | from | 125 | | |
Abstract
Y. Imai and K. Iseki introduced two classes of abstract algebras: BCK- algebras and BCIalgebras
[18-22]. It is known that the class of BCK-algebras is a proper subclass of the
class of BCI -algebras. In [3, 11, 16, and 17] Q. P. Hu and X. Li introduced a wide class
of abstract algebras: BCH - algebras. They have shown that the class of BCI- algebras is
a proper subclass of the class of BCH- algebras. In[58 ] the authors introduced the notion
of d-algebras, which is another useful generalization of BCK- algebras, and then they
investigated several relations between d-algebras and BCK- algebras as well as some
other interesting relations between d-algebras and oriented digraphs. Y.B. Jun, E. H. Roh
and H. S. Kim[26 ] introduce a new notion, called BH-algebras, which is a generalization
of BCH/ BCI /BCK-algebras. They also defined the notions of ideals in BH-algebras.
Recently J. Neggers and H. S. Kim[ 61 ] introduced the notion of B-algebra, and studied
some of its properties. In 1983, (Y. Komori) [39 ] introduced a notion of BCC-algebras,
and (W. A. Dudek) [15 ] redefined the notion of BCC-algebras by using a dual form of
the ordinary definition in the sense of Y. Komori. In [15 ], (W. A. Dudek and X. H.
Zhang) introduced a notion of BCC-ideals in BCC algebras and described connections
between such ideals and congruences. In 2001, (J.Neggers, S.S.Ahn and H.S.Kim) [ 59]
introduced a new notion, called a Q-algebra and generalized some theorems discussed in
BCI/BCK-algebras. Prabpayak and Leerawat [62, 63] introduced a new algebraic
structure which is called KU-algebra. They gave the concept of homomorphisms of KUalgebras
and investigated some related properties. The concept of a fuzzy set, was
introduced in [74]. O. Xi [72] applied the concept of fuzzy to BCK-algebras. In
[45,47,52 ], studied the fuzzification of BCK-algebra and BCI-algebra. In 2002, Mostafa
Abd-Elnaby and Yousef [55] introduced the notion of fuzzy ideals of KU-algebras and
then they investigated several basic properties which are related to fuzzy KU-ideals. They
described how to deal with the homomorphic image and inverse image of fuzzy KUideals.
They have also proved that the Cartesian product of fuzzy KU-ideals in Cartesian
product of fuzzy KU-algebras are fuzzy KU-idealsideals. Liu and Meng [23] introduced
the notion of sub-implicative ideals in BCI-algebras. Also Jun [43] introduced the notion
ii
of fuzzy sub-implicative ideals of BCI-algebras and obtained some related interesting
properties of these concepts.
Neutrosophic set and neutrosophic logic were introduced in1995 by Smarandache as
generalizations of fuzzy set and respectively intuitionistic fuzzy logic. In neutrosophic
logic, each proposition has a degree of truth (T), a degree of indeterminancy (I) and a
degree of falsity (F),where T,I, F are standard or non-standard subsets of ]−0, 1+[,
see [28 ,29,66,71].Neutrosophic logic has wide applications in science, engineering,
Information Technology, law, politics, economics, finance, econometrics, operations
research, optimization theory, game theory and simulation etc. Agboola and Davvaz
introduced the concept of neutrosophic BCI/BCK-algebras in [1, 2]. Davvaz B. Davvaz,
S M. Mostafa and F.Kareem [13] introduce a neutrosophic KU-algebra and KU-ideal and
investigate some related properties.Recently H.Wang et.al [71] introduced an instance of
neutrosophic set known as single valued neutrosophic set which was motivated from the
practical point of view and that can be used in real scientific and engineering
applications.
The thesis deals mainly with new algebraic structure is called sub-implicative ideals in
KU-algebras, fuzzy sub-implicative (implicative) ideals of KU-algebras , concepts single
valued neutrosophic of sub-implicative ideals in KU-algebras and investigate some
related properties.
This thesis has been mainly divided into five chapters. The main text of the thesis is in
chapters 1, 2, 3 and 4.
Chapter 0
In this chapter we have given an exhaustive of the basic definitions of some algebras and
fuzzy sets which are needed in the subsequent chapters and further, the history of the
problem.
Chapter 1
In this chapter, the notions of ku- sub implicative/ ku-positive/ ku- sub-commutative
ideals in KU-algebras are established. and some of their properties are investigated. Also,
iii
the relationships with ku -sub implicative ideals and ku- sub-commutative/ ku-positive
implicative are given.
Chapter 2
In this chapter, We consider the fuzzification of sub-implicative (sub-commutative)
ideals in KU-algebras, and investigate some related properties. We give conditions for a
fuzzy ideal to be a fuzzy sub-implicative(sub-commutative) ideal. We show that any
fuzzy sub-implicative (sub-commutative) ideal is a fuzzy ideal, but the converse is
not true. Using a level set of a fuzzy set in a KU-algebra, we give a characterization of
a fuzzy sub-implicative (sub-commutative) ideal.
.
Chapter 3
In this chapter, we consider ku - implicative ideal (briefly implicative ideal) in KUalgebras
.The notion of fuzzy implicative ideals in KU - algebras are introduced, several
appropriate examples are provided and their some properties are investigated. The image
and the inverse image of fuzzy implicative ideals in KU - algebras are defined and how
the image and the inverse image of fuzzy implicative ideals in KU - algebras become
fuzzy implicative ideals are studied. Moreover, the Cartesian product of fuzzy implicative
ideals in Cartesian product of KU – algebras are given.
Chapter 4
In this chapter, We consider the concepts single valued neutrosophic of sub-implicative
ideals in KU-algebras, and investigate some related properties. We give conditions for a
single valued neutrosophic ideal to be a single valued neutrosophic sub-implicative
ideal. We show that any single valued neutrosophic sub-implicative ideal is a single
valued neutrosophic ideal, but the converse is not true. Using a level set of a single
valued neutrosophic set in a KU-algebra, we give a characterization of single valued
neutrosophic sub-implicative ideal.