格代数和效应代数上的新的不确定性理论
摘要
本文研究了格上的凸模糊集理论和软集理论以及效应代数上的粗糙集理论.所作的具体工作如下：
(1)在过去几十年里,优化理论的发展使得模糊凸性在理论和应用方面遇到了问题,所以模糊凸性分析越来越受到了重视.受这一现状启发我们提出了广义凸模糊子格的概念,研究了一般凸模糊子格与两类广义凸模糊子格的等价性,用水平截集刻画了广义凸模糊子格,并且讨论了几类广义凸模糊子格的充要条件.同时,探讨了在同态映射f下,(λ,,μ)-凸模糊子格的像、逆像还是(λ,μ)-凸模糊子格；当μ和μ'分别是模糊子格λ和f(λ)的(∈,∈Vq)-模糊子格(或者理想)时,f(μ)是f(λ)的(∈,∈Vq)-模糊子格(或者理想),f-1(μ')是λ的(∈,∈Vq)-模糊子格(或者理想).最后,研究了(∈,∈Vqκ)-模糊子格(理想)的几个基本性质.
(2)对格上软集理论的研究具体有：首先,给出了软格的定义,讨论了软格在某些条件下,它通过交、并运算后还是软格；其次,用模糊点重新定义了软格,得到了∈-软格和q-软格,并且用广义模糊子格对它们进行了刻画；对软格和软理想模糊化后,得到了模糊软格和模糊软理想,我们主要讨论了它们的基本性质.最后,研究了(∈,∈Vq)-模糊软格及(∈,∈Vq)-模糊软理想,下面是重要定理中的其中两个：
(i)设(Fλ,A)和(Hμ,B)是L的两个(∈,∈Vq)-模糊软格,并且(F,A)是(Hμ,B)的(∈,∈Vq)-模糊软子格.如果f是从L到S的同态,则(f(Fλ),A)和(f(Hμ),B)都是S上的(∈,∈Vq)-模糊软格且(f(Fλ),A)是(f(Hμ),B)的(∈,∈Vq)-模糊软子格.
(ii)设(Fλ,A)是L的(∈,∈Vq)-模糊软格并且(α,I1),(βv,I2)是L上(Fλ,A)的(∈,∈Vq)-模糊软理想.如果I1和I2是不相交的,则(α,I1)∪(βv,I2)是(Fλ,A)的(∈,∈vq)-模糊软理想.
(3)粗糙集理论与应用的核心基础是从近似空间导出一对近似算子,即上近似算子和下近似算子(又称上、下近似集).我们试图以效应代数上的同余关系为基本要素,定义粗糙近似算子,从而导出粗糙集代数系统.然而,效应代数是个不完全代数,不是任意两个元素都能运算,所以我们在效应代数上赋予全运算从而构造了一个距离函数使得它与一类特殊理想,如Riesz理想,诱导的等价类有了很紧密的联系.随之,得到了粗糙近似算子及粗糙效应代数系统,并得出了其中的两个主要结论：
(i)设E是一个格序效应代数且(E,≤)是一个正交模格.并且设I是E的一个Riesz理想,X,Y是E的非空子集.则AprI,(X+Y)∈AprI(x)+AprI(Y).
(ii)设E是一个分配的格序效应代数,(E,≤)是一个正交模格.设I,J是E的两个Riesz理想并且X是E的非空子集.则AprI+J(X)(?) AprI(X)+AprJ(X).
这些工作为不确定性理论的进一步发展做出了一定的贡献.
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.
(1)在过去几十年里,优化理论的发展使得模糊凸性在理论和应用方面遇到了问题,所以模糊凸性分析越来越受到了重视.受这一现状启发我们提出了广义凸模糊子格的概念,研究了一般凸模糊子格与两类广义凸模糊子格的等价性,用水平截集刻画了广义凸模糊子格,并且讨论了几类广义凸模糊子格的充要条件.同时,探讨了在同态映射f下,(λ,,μ)-凸模糊子格的像、逆像还是(λ,μ)-凸模糊子格；当μ和μ'分别是模糊子格λ和f(λ)的(∈,∈Vq)-模糊子格(或者理想)时,f(μ)是f(λ)的(∈,∈Vq)-模糊子格(或者理想),f-1(μ')是λ的(∈,∈Vq)-模糊子格(或者理想).最后,研究了(∈,∈Vqκ)-模糊子格(理想)的几个基本性质.
(2)对格上软集理论的研究具体有：首先,给出了软格的定义,讨论了软格在某些条件下,它通过交、并运算后还是软格；其次,用模糊点重新定义了软格,得到了∈-软格和q-软格,并且用广义模糊子格对它们进行了刻画；对软格和软理想模糊化后,得到了模糊软格和模糊软理想,我们主要讨论了它们的基本性质.最后,研究了(∈,∈Vq)-模糊软格及(∈,∈Vq)-模糊软理想,下面是重要定理中的其中两个：
(i)设(Fλ,A)和(Hμ,B)是L的两个(∈,∈Vq)-模糊软格,并且(F,A)是(Hμ,B)的(∈,∈Vq)-模糊软子格.如果f是从L到S的同态,则(f(Fλ),A)和(f(Hμ),B)都是S上的(∈,∈Vq)-模糊软格且(f(Fλ),A)是(f(Hμ),B)的(∈,∈Vq)-模糊软子格.
(ii)设(Fλ,A)是L的(∈,∈Vq)-模糊软格并且(α,I1),(βv,I2)是L上(Fλ,A)的(∈,∈Vq)-模糊软理想.如果I1和I2是不相交的,则(α,I1)∪(βv,I2)是(Fλ,A)的(∈,∈vq)-模糊软理想.
(3)粗糙集理论与应用的核心基础是从近似空间导出一对近似算子,即上近似算子和下近似算子(又称上、下近似集).我们试图以效应代数上的同余关系为基本要素,定义粗糙近似算子,从而导出粗糙集代数系统.然而,效应代数是个不完全代数,不是任意两个元素都能运算,所以我们在效应代数上赋予全运算从而构造了一个距离函数使得它与一类特殊理想,如Riesz理想,诱导的等价类有了很紧密的联系.随之,得到了粗糙近似算子及粗糙效应代数系统,并得出了其中的两个主要结论：
(i)设E是一个格序效应代数且(E,≤)是一个正交模格.并且设I是E的一个Riesz理想,X,Y是E的非空子集.则AprI,(X+Y)∈AprI(x)+AprI(Y).
(ii)设E是一个分配的格序效应代数,(E,≤)是一个正交模格.设I,J是E的两个Riesz理想并且X是E的非空子集.则AprI+J(X)(?) AprI(X)+AprJ(X).
这些工作为不确定性理论的进一步发展做出了一定的贡献.
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.
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