الفهرس 
Important Concepts on Fuzzy setsOnly 14 pages are availabe for public view
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Abstract
In 1966,Y.Imai and K.Isèki introduced two classes of abstract algebras: BCKalgebras
and BCI-algebras see[47,48,50,56]It is known that the class of BCKalgebras
is a proper subclass of the class of BCI-algebras. Neggers et al [122]
introduced a notions, called Q-algebras, which is a generalization of BCH /
BCI/BCK-algebras and generalized some theorems discussed in BCIalgebras.
Moreover, Ahn and Kim [6] introduced the notions of QS-algebras
which is a proper subclass of Q-algebras. Kondo [97] proved that, each
theorem of QS-algebras is provable in the theory of Abelian groups and
conversely each theorem of Abelian groups is provable in the theory of
QS-algebras. QS algebra in the fuzzy setting have also been considered by
many authors .
The concept of fuzzy sets was introduced by Zadeh [152]. In 1991, Xi
[149,150] applied the concept of fuzzy sets to BCI, BCK -algebras. Since its
inception, the theory of fuzzy sets , ideal theory and its fuzzification has been
developed in many directions and applied to a wide variety of fields. Jun et al.
[64,65,66,70] introduced the notion of cubic sub-algebras/ideals in BCK/BCIalgebras,
and then they investigated several properties. They discussed the
relationship between a cubic sub-algebra and a cubic ideal. Also, they
provided characterizations of a cubic sub-algebra/ideal and considered a
method to produce a new cubic subalgebra from an old one. Lee [93]
introduced an extension of fuzzy sets named bipolar-valued fuzzy sets.
Bipolar-valued fuzzy sets are an extension of fuzzy sets whose membership
degree range is enlarged from the interval [0, 1] to [-1, 1]. Recently Jun et al.
[71] introduced a new function which is called negative-valued function, and
constructed N-structures. They applied N-structures to BCK/BCI-algebras,
and discussed N-subalgebras and N-ideals in BCK/BCI-algebras. Jun et al.
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[72] established an extension of a bipolar-valued fuzzy set, which is
introduced by Lee [100]. They called it a crossing cubic structure, and
investigated several properties. They applied crossing cubic structures to
BCK/BCI-algebras, and studied crossing cubic sub algebras.
The thesis deals mainly with new algebraic structure which is called Crossing
cubic ideals of some algebras ,the concepts crossing cubic QS- ideal of QSalgebra
were introduced and some related properties were investigated.
This thesis has been mainly divided into four chapters. The main text of the
thesis is in chapters 1, 2,3 and 4.
Chapter 1 is called: “Introduction to fuzzy sets and its extension”
In this chapter, we begin with basic crisp sets (classical sets) and fuzzy sets
definitions. We consider relation between universal set ,fuzzy set, 𝛼- cut set
and level set. At the end of this chapter ,we show some types of fuzzy sets
such as intuitionistic fuzzy sets, interval-valued fuzzy sets, bipolar-valued
fuzzy set and provide operations on them including examples .
Chapter 2 is called: “Review on BCI/BCK-Algebras and Development”.
In this chapter we have given an exhaustive of the basic definitions of some
algebras which are needed in the subsequent chapters and. We begin with
basic BCK-algebra theory including several examples. We consider sub -
algebras, bounded BCK-algebras, positive implicative BCK-algebras,
commutative BCK-algebras, implicative BCK-algebras, BCK-algebras with
condition (S). we provide characterizations of commutative, positive
implicative and implicative BCK-algebras.
The ideal theory plays an important role for the general development of BCKalgebras.
We discuss ideals , implicative ideals, positive implicative ideals,
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commutative ideals, maximal ideals, and we give basic properties and some
characterizations of such ideals. at the end of this chapter the homomorphic
image ( preimage) of some-ideals of some algebras under homomorprhism of
algebras are discussed . Finally, many related results have been derived. We
review several classes of abstract algebras related to BCK/BCI ” we begin
with basic BCH / BCI / BG / BF -algebras theorems including several
examples. We then consider sub algebras in each algebras.
Chapter 3 is called “Introduction to QS -algebras and its related topic”
In this chapter ,we list some algebras related to Algebra QS through some
researches studies. we begin with basic QS-algebra theory. We give relations
between QS-algebras and the different algebras (BP/BOI/BM-algebras). We
study QS-ideals in QS-algebras, relation between (BCK / QS-ideals) in a Qalgebra,
(QS / BCK-ideals) in BCK-algebra and homomorphism of QSalgebra.
We introduce the notion of fuzzy QS-ideal of QS-algebra as
generalization of fuzzy ideal of QS-algebra and then we investigate several
basic properties which are related to fuzzy QS-ideals. We investigate how to
deal with the homomorphic image and inverse image of fuzzy QS-ideal.
We show that if 𝜇 and 𝛽 are fuzzy QS-ideals of QS-algebras X, then 𝜇 × 𝛽 is
a fuzzy QS-ideal of 𝑋 × 𝑋 conversely, we show that if 𝜇 × 𝛽 is a fuzzy QSideal
of 𝑋 × 𝑋, either or is a fuzzy QS-ideal of a QS-algebra X at the
end of this chapter we give example shows that union of two QS-ideals may
not QS-ideal and the union of two fuzzy QS-ideals may QS-ideals.
Chapter 4 is called: (𝜶̃ . 𝜶)- crossing cubic QS-ideal of QS-algebras
In this chapter, As an extension of bipolar-valued fuzzy sets, the notions of
(𝛼̃ . 𝛼)- crossing cubic QS-ideal of QS-algebras are introduced, and several
related properties are investigated. characterizations of (𝛼̃ . 𝛼)- crossing cubic
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QS-ideal on QS-algebras are established. The relations between ( 𝛼̃ . 𝛼 )-
crossing cubic subalgebras and ( 𝛼̃ . 𝛼 )- crossing cubic QS-ideal of QSalgebras
are investigated. Moreover, the homomorphic image ( pre image) of
(𝛼̃ . 𝛼)- crossing cubic QS-ideal of a QS-algebra under homomorprhism of a
QS-algebras is discussed