New Theories Of Uncertainty On Lattice And Effect Algebras

In this paper, we study the convex fuzzy set theory and soft set theory on lattice and the rough set theory on effect algebra. The specific work we have done is as follows:(1) In the past few decades, the development of optimize theory makes fuzzy convexity encountered problem in theoretical and applied aspects. The fuzzy convex analysis is paid more and more attention. Inspired by this situation, we propose the concept of generalized convex fuzzy sublattices, and study the equivalence of convex fuzzy sublattices with two classes of generalized convex fuzzy sublattice. Also we portray generalized convex fuzzy sublattices by level sets and discuss necessary and sufficient conditions of some types generalized convex fuzzy sublattices. Furthermore, we explore that under homomorphisms f, the image and inverse image of (λ,μ)-convex fuzzy sublattice are still (λ,μ)-convex fuzzy sublattice. When μ and μ’are (∈,∈Vq)-fuzzy sublattices(resp. ideals) of fuzzy sublattices λ and f(λ) respectively, then f(μ) is (∈,∈Vq)-fuzzy sublattices(resp:ideals) of(λ), f-1(μ’) is (∈,∈Vq)-fuzzy sublattices (resp. ideals) of λ. Finally, we study some fundamental properties of (∈,∈∨qk)-fuzzy lattices (ideals).(2) The specific research on soft set theory is as following:First, the definition of soft lattice is given. And the intersection, and union on them are still soft lattice. Second, soft lattice is redefined by fuzzy point and∈-soft lattice and q-soft lattice are obtained. Also they are portrayed by generalized fuzzy lattice. Fuzzy soft lattice and fuzzy soft ideal are obtained by fuzzifi-cation the soft lattice and soft ideal respectively, and their basic properties are discussed. Finally,(∈,∈∨g)-fuzzy soft lattice and (∈,∈∨q)-fuzzy soft ideal are studied. The following are two of important theorems:(i) Let (Fλ,A) and (Hμ,B) be two (G, G Vg)-fuzzy soft lattices over L, and (Fλ, A) be a (∈,∈Vq)-Fuzzy soft sublattice of (Hμ,B). If F is a homomorphism from L to S, then (f(Fλ),A) and (f(Hμ),B) are both (∈,∈Vg)-fuzzy soft lattices over S and (f(Fλ),A) is a (∈,∈Vg)-fuzzy soft sublattice of (f(Hμ), B).(ii) Let(Fλ,A) be a (∈,∈Vg)-fuzzy soft lattice over L and (αμ,Ii),(βv,I2) be (∈,∈Vq)-fuzzy soft ideals of (Fλ,A) over L. If I1and I2are disjoint, then (αμ,I1)∪(βv,I2) is a (∈,∈Vq)-fuzzy soft ideal of (F\,A).(3) The core foundation of rough set theory and application is a pair of approximation operators induced from the approximation space, namely the upper approximation operator and lower approximation operator (also called upper and lower approximation set). We try to define rough approximation operators by the basic elements of congruence relations on effect algebra, thus induce rough set algebraic system. However, the effect algebra is incomplete algebra, not any two elements can computing, so we give full operation in constructive method on effect algebra to construct a distance function making it have a very close link with a special kind of ideal, such as Riesz ideal which induced equivalence class. Then, we obtain rough approximation operators, rough effect algebra system. The two of main results are as following:(i) Let I be a Riesz ideal of E and (E,≤) be orthomodular lattice. And let X, Y be non-empty subsets of E, Then Apγi(X+Y) C AprI(X)+AprI(Y).(ii) Let I, J be two ideals of E and (E,≤) be orthomodular lattice. And let X be a non-empty subset of E. Then ApγI+J(X) C AprI(X)+ApγJ(X). These work has made a certain contribution to the further development of the uncertainty theory.