The Fuzzy Theory Of Some Classes Of Algebraic Structures

The research on fuzzy algebra has become one of the most active topics in fuzzy mathematics in recent years. Since Rosenfeld [50] introduced the fuzzy subsets into the realm of group theory, many mathematicians have been involved in extending the concepts and results of abstract algebra to the broader framework of the fuzzy setting. The study of fuzzy algebra relates to various aspects of algebra, such as fuzzy group, fuzzy ring, fuzzy module, fuzzy homology and so on. One of the basic principles of such fuzzy subsystems is that a fuzzy subsetμof an algebraic system R is a fuzzy subsystem of R if and only if the level subsetsμ_t = {x∈R,μ(x)≥t, for 0≤t≤1} are subsystems of R. Thereout, we can study the subsystems of R by fuzzy subsystems. and also characterize the fuzzy subsets by the subsystems of R. Following from this basic principle, Lie superalgebras are introduced into fuzzy subsets and the framework of fuzzy Lie theory are established. Considering Hopf algebras and quantum groups are the generalizations of Lie theory, we introduce the coalgebras into fuzzy subsets, and attempt to establish the theory of fuzzy quantum groups. Moreover, comodules as new objects are also introduced in this thesis, we can discuss fuzzy comoduels' properties.The main results of this thesis are presenting the fuzzy theory of Lie superalgebras. studying the properties of fuzzy Lie sub-superalgebras and fuzzy ideals in Lie supe; -algebras; presenting the definition of fuzzy subcoalgebras, studying the properties of fuzzy subcoalgebras. fuzzy (left/right) coideals, discussing the relations between fuzzy subcoalgebras and fuzzy subalgebras as well as between fuzzy (left/right) coideals and fuzzy (left/right) ideals; introducing the comodules into fuzzy theory, characterizing the properties of fuzzy subcomodules.In Chapter one firstly we provide a brief introduction of the background and the recent situation of development and research of fuzzy algebras. We also show that our work is inevitability and feasibility in this thesis. Secondly we introduce some basic properties of fuzzy subsets and the Zadeh's extension principle.In Chapter two firstly according to Z₂-graded vector spaces, we define Z₂-graded fuzzy vector subspaces. On this basis, the concepts of fuzzy Lie sub-superalgebras and fuzzy ideals are presented, the characterization of Theorem 2.2.6 is obtained: A fuzzy subsetμof Lie superalgebra (?) is a fuzzy Lie sub-superalgebra(resp. fuzzy ideal) of (?) if and only if the level subsetsμ_t = {x∈(?) :μ(x)≥t for 0≤t≤μ(0)} are Lie sub-superalgebras (resp. ideals) of (?). Studying the basic properties of fuzzy Lie sub-superalgebras and fuzzy ideals, we can obtain that fuzzy Lie sub-superalgebras and fuzzy ideals are the generalizations of fuzzy subalgebras and fuzzy ideals of Lie algebras.Secondly the fuzzy-quotient Lie superalgebra is defined using fuzzy ideal, and the conclude which fuzzy-quotient Lie superalgebra is Lie superalgebra is proved. So, the fuzzy subset of fuzzy-quotient Lie superalgebra can be studied, and the conclusion of Theorem 2.3.7 is obtained. After considering the fuzzy theory of quotient Lie superal-gebras, Theorem 2.4.1 can be obtained. The fuzzy ideals of fuzzy Lie sub-superalgebras are defined, the properties of this kind of fuzzy ideals are also studied and Theorem 2.4.3, Theorem 2.4.4, Theorem 2.4.5 and Theorem 2.4.6 are obtained.The homomorphisms of Lie superalgebras are one of the basic definitions between Lie superalgebras. The main work of the section five of Chapter two is that some fundamental concepts of fuzzy sub-superalgebras and fuzzy ideals under homomorphisms are discussed. The main results include Proposition 2.5.2, Proposition 2.5.3 and Proposition 2.5.4. Combined with the work of the section four, Theorem 2.5.7, Theorem 2.5.8. Theorem 2.5.9 and Theorem 2.5.10 are obtained further.Lastly the solvable fuzzy ideals and the nilpotent fuzzy ideals are studied. At first. the sup-min product [ , ] of Lie superalgebras is defined. Yehia [58] proved that in the case of Lie algebras, sup-min product satisfied [μ,ν₁ +ν₂] (?) [μ,ν₁]+[μ,ν₂]. We prove in Theorem 2.6.4 that sup-min product [ , ] keeps bilinear. It is also true in Lie algebras.Theorem 2.6.4 Letμ₁,μ₂,ν₁,ν₂ andμ,νbe fuzzy vector subspaces of (?). Then for anyα,β∈k, we have In order to define the solvable fuzzy ideals and the nilpotent fuzzy ideals, Theorem 2.6.8 is proved that fuzzy ideals are closed with regard to sup-min product.Theorem 2.6.8 Letμ,νbe any two fuzzy ideals of (?). Then [μ,ν] is also a fuzzy ideal of (?).At the end of the section six, the solvable fuzzy ideals and the nilpotent fuzzy ideals are defined, some properties of the solvable fuzzy ideals and nilpotent fuzzy ideals are studied, and the relation between the solvable fuzzy ideals of Lie superalgebras and the solvable fuzzy ideals of Lie superalgebras' even parts is characterized. Moreover, under the condition of fuzzy super-Jacobi identity, the set of the fuzzy ideals of Lie superalgebras is Lie superalgebra. The main results include Proposition 2.6.13, Theorem 2.6.15, Theorem 2.6.16, Theorem 2.6.19 and Theorem 2.6.20.In Chapter three firstly we require that the decomposition of A(c) is unique. Under this condition, fuzzy subcoalgebras, fuzzy (left/right) coideals are defined. The equivalent characterization of fuzzy subcoalgebras is given by using level subsets and strong level subsets, we obtain Theorem 3.2.3. The corresponding equivalent definitions of fuzzy (left/right) coideals are Theorem 3.2.7 and Theorem 3.2.11. The result of the Note 3.2.8(2) is more easily to get than coalgebras.Secondly the properties of fuzzy subcoalgebras and fuzzy (left/right)coideais are discussed under coalgebra homomorphisms. Proposition 3.3.2 and Proposition 3.3.3 are obtained.Lastly the duality between fuzzy subcoalgebras and fuzzy ideals as well as the duality between fuzzy coideals and fuzzy subalgebras is studied. For the latter, infinitedimensional cases and finite-dimensional cases are discussed respectively. The main results are as follows:Proposition 3.4.3 (1) Letμbe a fuzzy subcoalgebra of C. Thenμ^* is a fuzzy ideal of C^*.(2) Letμbe a fuzzy coideal of C. Thenμ^* is a fuzzy subalgebra of C^*.(3) Letμbe a fuzzy left (right) coideal of C. Thenμ^* is a fuzzy left (right) coideal of C^*. Proposition 3.4.4 (1) Letμbe fuzzy ideal of finite dimensional algebra A. Thenμ^* is fuzzy subcoalgebra of A^*.(2) Letμbe fuzzy left (right) ideal of finite dimensional algebra A. Thenμ^* is fuzzy left (right) coideal of A^*.Proposition 3.4.7(1) Letμbe a fuzzy ideal of A. Thenμ^°is a fuzzy subcoalgebra of A^°.(2) Letμbe a fuzzy left (right) ideal of A. Thenμ^°is a fuzzy left (right) coideal of A^°.As an application of the above study, fuzzy morphisms are discussed and Proposition 3.4.11 and Proposition 3.4.12 are obtained.In Chapter four firstly fuzzy subcomodules are defined on the basis of adjusting the definition of fuzzy submodules, and the properties of fuzzy subcomodules are studied. Theorem 4.2.5 and Theorem 4.2.6 and Theorem 4.2.7 are obtained.Secondly the properties of fuzzy subcomodules are discussed under comodule homomorphisms . Theorem 4.3.3 and Theorem 4.3.4 are obtained.Lastly projective fuzzy subcomodules and injective fuzzy subcomodules are denned and the necessary and sufficient conditions are studied. The main results are as follows:Theorem 4.4.3 (1) (P,μ) is projective fuzzy subcomodule if and only if P is projective comodule andμ= l₀.(2) (I,μ) is injective fuzzy subcomodule if and only if I is injective comodule andμ= 1.(1(x) = 1, for any x∈I.)...