Novikov Algebras,L-Valued Categories And Their Applications

There are three parts in this thesis.Part 1 devotes to Novikov algebras structures and classifications.First of all,we construct two kinds of infinite-dimensional Novikov algebras and give their realization,respectively.Moreover,we studies quasiderivations and quasicentroids of Novikov algebras,and we determine the dimension of Novikov algebras A with condition QDer(A)=End(A).We also show that the quasiderivations of A can be embedded as derivations in a larger Novikov algebra.Part 2 studies L-valued categories structures of L-fuzzy sets category and L-fuzzy left R-modules category,where L denotes a complete Heyting algebra.First,we prove that the category Set(L)of L-fuzzy sets is isomorphic to the category Set(f L).Then,we discuss several fuzzy sets with different truth-value sets,and point out that they are special fuzzy theories essentially.We establish one to one correspondence between the category of fuzzy theory and the Kleisli ca.tegory which is constructed by monad induced by the fuzzy theory.Furthermore,we give the definition of L-fuzzy left R-modules limit(colimit)by virtue of pointwise and pointless depiction,discuss the limits(colimits)of existence,uniqueness and structural theorem,and get limit functor and constant direct system functor is an adjoint pair.Finally,we define functors L-ext and L-Ext,and get the isomorphism relations between them.Part 3 dedicates to the application of L-valued categories to Novikov algebras.First,we introduce the concepts of L-fuzzy pre-Lie subalgebra,L-fuzzy Novikov subalgebra and L-fuzzy sub-adjacent Lie algebra,and discussed the L-fuzzy structures of some classical pre-Lie algebras and Novikov algebras.Furthermore,we apply the concept of L-fuzzy sets to Novikov algebras,and introduce the concepts of L-fuzzy ideals and L-fuzzy subalgebras.We get a sufficient and neccessary condition such that an L-fuzzy subspace is an L-fuzzy ideal.We show that the quotient algebra A/? of the L-fuzzy ideal is isomorphic to the algebra A/A? of the non-fuzzy ideal A?.