幂线性空间
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摘要
随着模糊数学的发展,集值映射的重要性的日益突出,各种数学结构都有由论域向其幂集上提升的需要。自从李洪兴教授在文献[1]中考虑了代数结构的提升问题,并首次提出了HX群的概念,文献[2]提出代数群的提升—幂群的概念以来,超代数的研究引起了不少学者和爱好者的关注,文献[3]-[23]得到了幂群的一系列很好的结果,文献[24]-[38]给出了环的幂集提升—幂环,并得到了幂环的一系列性质,文献[39]-[43]给出了格的幂集提升—幂格,并相应得到一系列有价值的成果。超代数结构的研究通常的研究方法:(1)代数运算的提升;(2)幂代数结构概念及其实例;(3)幂代数结构的性质。
首先线性空间上的线性运算的幂集提升
设(V,F,+,·)是一个线性空间,在P*(V)(?)P(V)-Φ中定义二元集运算(称之为二元幂加法):
A(?)B={a+b|a∈A,b∈B};二元幂数量乘积运算(称之为二元幂数量乘积):
k(?)A={ka|a∈A},其中k∈F,A,B∈P*(V)。
定义2.3.1设(V,F,+,·)是一个线性空间,在P*(V)(?)P(V)-Φ中定义二元幂加法
A(?)B={a+b|a∈A,b∈B};二元幂数量乘积运算:
k(?)A={ka|a∈A},其中k∈F,A,B∈P*(V),如果P*(V)关于以上两个运算构成数域F上的线性空间,那么称P*(V)是由数域F上线性空间V诱导出的一个自然幂线性空间。
定理2.3.1自然幂线性空间一定是原线性空间或其某个子空间。
定义2.3.2设(V,F,+,·)是一个线性空间,在P*(V)(?)P(V)-Φ中定义二元幂加法
A(?)B={a+b|a∈A,b∈B};二元幂数量乘积运算:
k(?)A={ka|a∈A}(k≠0,特别地当k=0时,规定0(?)A=0),其中k∈F,A,B∈P*(V),如果P*(V)关于以上两个运算构成数域F上的线性空间,那么称P*(V)是由数域F上线性空间V诱导出的一个幂线性空间。
自然幂线性空间和幂线性空间统称为幂线性空间。
自然幂线性空间必是幂线性空间的特例。
定理2.3.2幂线性空间P*(V)一定是原线性空间V或其子空间的一个商空间。
定义2.4.1设(P*(V),F,(?),(?))是由线性空间(V,F,+,·)诱导出的一个幂线性空间,且A,A_1,A_2,…,A_s∈P*(V),若(?)k_1,k_2,…,k_s∈F,使得:
k_1A_1+k_2A_2+…+k_sA_s=A则称A可由幂向量组A_1,A_2,…,A_s幂线性表出,k_1A_1+k_2A_2+…+k_sA_s称为幂向量组A_1,A_2,…,A_s幂线性组合。
定义2.4.2设(P*(V),F,(?),(?))是由数域F上n维线性空间(V,F,+,·)诱导出的一个幂线性空间,且A_1,A_2,…,A_s∈P*(V),若(?)k_1,k_2,…,k_s∈F不全为零,使得:
k_1A_1+k_2A_2+…+k_sA_s=0则称幂向量组A_1,A_2,…,A_s是幂线性相关的;否则称为幂线性无关。
定义2.4.3设(P*(V),F,(?),(?))是由数域F上n维线性空间(V,F,+,·)诱导出的一个幂线性空间,且A_1,A_2,…,A_m∈P*(V),若满足:
(1)A_1,A_2,…,A_m是幂线性无关的;
(2)(?)A∈P*(V)都可由幂向量组A_1,A_2,…,A_m幂线性表出则称幂向量组A_1,A_2,…,A_m是幂线性空间(P*(V),F,(?),(?))的一组幂基,简称为基,并称幂线性空间(P*(V),F,(?),(?))是m维的,记为:dim(P*(V))=m。
定理2.4.1幂线性空间的维数等于线性空间或其子空间的维数与幂零元的维数之差。
定理2.4.2自然幂线性空间的维数等于线性空间或其子空间的维数。
定义2.5.1设(P*(V),F,(?),(?))数域F上n维线性空间V的幂线性空间,W为P*(V)的非空子集,若(W,F,(?),(?))构成数域F上的幂线性空间,则称W为P*(V)的幂线性子空间,简称幂子空间。
定义2.5.2设W_1,W_2是(P*(V),F,(?),(?))的幂线性子空间,则称W_1∩W_2为P*(V)的幂子空间W_1与W_2的幂交。
定义2.5.3设W_1,W_2是(P*(V),F,(?),(?))的幂线性子空间,则称W_1+W_2为P*(V)的幂子空间W_1与W_2的幂和。
定理2.5.1设W_1,W_2是(P*(V),F,(?),(?))的幂子线性空间,则W_1∩W_2也是P*(V)的幂子空间,即:幂子空间的的幂交仍为幂子空间。
定理2.5.2设W_1,W_2是(P*(V),F,(?),(?))的幂子线性空间,则W_1+W_2也是P*(V)的幂子空间,即幂子空间的的幂和仍为幂子空间。
定理2.5.3设W_1,W_2是(P*(V),F,(?),(?))的幂子线性空间,则
dim(W_1)+dim(W_2)=dim(W_1∩W_2)+dim(W_1+W_2)。(此为幂线性空间的维数公式。)
定义2.6.1设(P*_1(V),F,(?)_1,(?)_1)与(P*_2(W),F,(?)_2,(?)_2)分别是线性空间(V,F,+_1,·_1)与(W,F,+_2,·_2)的幂线性空间,如果P*_1(V)到P*_2(W)的一个线性映射σ满足:
(1)σ是满射;
(2)σ(A_1(?)_1A_2)=σ(A_1)(?)_2σ(A_2);
(3)σ(k(?)_1A_1)=k(?)_2σ(A_1)其中A_1,A_2∈P*_1(V),k∈F,则σ称是P*_1(V)到P*_2(W)的同态映射,也称P*_1(V)与P*_2(W)是同态的。
定义2.6.2设(P*_1(V),F,(?)_1,(?)_1)与(P*_2(W),F,(?)_2,(?)_2)分别是线性空间(V,F,+_1,·_1)与(W,F,+_2,·_2)的幂线性空间,如果P*_1(V)到P*_2(W)的一个线性映射σ满足:
(1)σ是双射;
(2)σ(A_1(?)_1A_2)=σ(A_1)(?)_2σ(A_2);
(3)σ(k(?)_1A_1)=k(?)_2σ(A_1)其中A_1,A_2∈P*_1(V),k∈F,则σ称是P*_1(V)到P*_2(W)的同构映射,也称P*_1(V)与P*_2(W)是同构的。
定理2.6.1同一个数域下的两个有限维的幂线性空间同构当且仅当它们维数是相同的。
定理2.6.2任意线性空间与它诱导出的幂线性空间同态。
定理2.6.3任意幂线性空间都与其原线性空间的某个子空间同构。
推论同一个数域下的任意两个线性空间或者幂线性空间的维数相等,则它们必同构。
对线性空间的线性运算的广义幂集提升,一般集合上定义幂集线性运算,设(V,F,+,·)是一个线性空间,在P*(V)(?)P(V)-Φ中定义二元集运算(称之为二元幂加法):
A(?)B=C∈P*(V);二元幂数集乘积运算:
k(?)A=D∈P*(V),其中k∈F,A,B∈P*(V)。
定义3.2.1设(V,F,+,·)是一个线性空间,在P*(V)(?)P(V)-Φ中定义二元幂加法和幂数量乘积运算,使得:A(?)B=C∈P*(V),k(?)A=D∈P*(V),其中k∈F,A,B∈P*(V),如果P*(V)关于以上两个运算构成数域F上的线性空间,则称P*(V)为数域F上线性空间V的第一广义幂线性空间,简称V上的第一广义幂线性空间,记为:(P*(V),F,(?),(?))_1。
定义3.2.2设X是非空集合,F是数域,在P*(X)(?)P(X)-Φ与F中定义二元集加法和幂数集乘积运算:A(?)B∈P*(X),k(?)A∈P*(X),其中k∈F,A,B∈P*(X),若P*(X)关于以上两个运算构成数域F上的线性空间,则称P*(X)为数域F上线性空间V的第二广义幂线性空间,简称V上的第二广义幂线性空间,一般也称为幂集线性空间,记为:(P*(X),F,(?),(?))_2。
第一广义幂线性空间与第二广义幂线性空间通称为广义幂线性空间。为方便起见,统一记广义幂线性空间为(P*(X),F,(?),(?))。
相应的可以定义广义幂线性空间上的幂线性表出、幂线性相关、幂线性无关、幂基、维数、广义幂子空间、同态与同构以及系列性质。
运算的提升可以得出各种超结构,如幂群、幂环、幂格、幂模等,当然也包括幂线性空间。尽管提升后的超结构一般来讲并未脱离原有结构的范畴,但却拓宽了相应的内涵和应用范围。
With the development of fuzzy math,the importance of set-valued function is growing,it need to upgrade that Various mathematical structures from the universe to power set.Since Li Hongxing professor in the literature[1]consider the upgrade of the algebraic structure,and the concept of the HX- group is given at first,upgrading of algebraic group- power group is discussed in literature[2].More and more scholars and enthusiasts concern studying of super algebra,A series of very good results of power group is given in the literature[3]-[23],The power set upgrade of the ring-power ring is discussed in the literature[24]-[38],and The power set upgrade of lattic-power lattic is discussed in the literature[39]-[43],and a series of valuable results is given,the usual methods of studying super-algebra structure:(1)algebra operators upgrade;(2) the concept and examples of power algebraic structure;(3)the nature of power algebraic structure.
First,in this paper,power set upgrade of linear operators is discussed in linear space.Let(V,F,+,·)is a linear space,definition of the power set operator is Given in P*(V)(?)P(V)-Φ:
A(?)B={a+b|a∈A,b∈B}; definition of the power set operator is Given in P×P*(V):
k(?)A={ka|a∈A}, where k∈F,A,B∈P*(V).
Definitions 2.3.1 Let(V,F,+,·)is a linear space,definition of the power set operator is given in P*(V)(?)P(V)-Φ:
A(?)B={a+b|a∈A,b∈B}; definition of the power set operator is Given in P×P*(V):
k(?)A={ka|a∈A}, where k∈F,A,B∈P*(V).If(P*(V),F,(?),(?))is linear space with two operations, then We call that it is a natural power linear space in number field F.
Theorem 2.3.1 natural power linear space is the original linear space or a subspace of the original linear space.
Definitions 2.3.2 Let(V,F,+,·)is a linear space,definition of the power set operator is Given in P*(V)(?)P(V)-Φ:
A(?)B={a+b|a∈A,b∈B}; definition of the power set operator is Given in P×P*(V):
k(?)A={ka|a∈A}, where k(≠0)∈F(Especially,if k=0,0(?)A=0),A,B∈P*(V).If(P*(V),F,(?),(?))is linear space with two operations,then We call that it is a power linear space in number field F.
Natural power linear space and power linear space space power collectively referred to as power linear space.
It is a special case that natural power linear space is power linear space.
Theorem 2.3.2 power linear space is a quotient space of the original linear space or it's sub-space.
Definitions 2.4.1 Let(P*(V),F,(?),(?))is a power linear space,and
A,A_1,A_2,...,A_s∈P*(V), If(?)k_1,k_2,...,k_s∈F,such that:
k_1A_1+k_2A_2+...+k_sA_s=A, then A is claimed power-linear presentation by the power vector group A_1,A_2,...,A_s, In this case,k_1A_1+k_2A_2+...+k_sA_2≤is claimed power linear combination of power vector group A_1,A_2,...,A_s.
Definition 2.4.2 Let(P*(V),F,(?),(?))is a power linear space,and
A_1,A_2,...,A_s∈P*(V), If(?)k_1,k_2,...,k_s∈F,and k_1≠0 or k_2≠0 or...k_s≠0,such that:
k_1A_1+k_2A_2+...+k_sA_s=0, then the power vector group A_1,A_2,...,A_s is claimed power linear dependent, otherwise,known as the power linearly independent.
Definition 2.4.3 Let(P*(V),F,(?),(?))is a power linear space,and
A_1,A_2,...,A_m∈P*(V), and satisfy that:
(1)A_1,A_2,...,A_s is power linearly independent;
(2)(?)A∈P*(V),all is the power linear combination of A_1,A_2,...,A_s, then A_1,A_2,...,A_s is claimed a power base in(P*(V),F,(?),(?)).In this base,saying power linear space(P*(V),F,(?),(?))is m-dimensional linear space.
Theorem 2.4.1 The dimension of power linear space equivalent to the poor of the dimension of the original linear space and the dimension of O,or the poor of the dimension of sub-space of the original linear space and the dimension of O.
Theorem 2.4.2 The dimension of natural power linear space equivalent to the dimension of the original linear space or dimension of sub-space of the original linear space.
Definitions 2.5.1 Let(P*(V),F,(?),(?))is a power linear space,and W(?)P*(V),If W is a power linear space,then We call that W is a power linear sub-space of P*(V).
Definitions 2.5.2 Let W_1,W_2 is a power linear sub-space of(P*(V),F,(?),(?)), then We call that W_1∩W_2 is the power intersection of power sub-space W_1 and W_2.
Theorem 2.5.1 Let W_1,W_2 is a power linear sub-space of(P*(V),F,(?),(?)),then it is power linear sub-space that the power intersection of W_1 and W_2.
Theorem 2.5.3 Let W_1,W_2 is a power linear sub-space of(P*(V),F,(?),(?)),then
dim(W_1)+dim(W_2)=dim(W_1∩W_2)+dim(W_1+W_2)。
Definitions 2.6.1 Let(P*_1(V),F,(?)_1,(?)_1)and(P*2(W),F,(?)_2,(?)_2)are two power linear space,σis a linear map for P*_1(V)to P*_2(W),if satisfy:
(1)σIs Surjectivity;
(2)σ(A_1,(?)_1,A_2)=σ(A_1)(?)_2σ(A_2);
(3)σ(k(?)_1A_1)=k(?)_2σ(A_1), where A_1,A_2∈P*_1(V),k∈F,then,σis claimed to homomorphism mapping for P*_1(V)to P*_2(W).
Definitions 2.6.2 Let(P*_1(V),F,(?)_1,(?)_1)and(P*_2(W),F,(?)_2,(?)_2)are two power linear space,σis a linear map for P*_1(V)to P*_2(W),if satisfy:
(1)σBijective;
(2)σ(A_1(?)_1A_2)=σ(A_1)(?)_2σ(A_2);
(3)σ(k(?)_1A_1)=k(?)_2σ(A_1) where A_1,A_2∈P*_1(V),k∈F,then,σis claimed to isomorphism mapping for P*_1(V)to P*_2(W).
Theorem 2.6.1 Under same number field,the two finite dimensional power linear space is isomorphism if and only if their dimension is the same.
Then,generalized power set upgrade of linear operators is discussed in linear space. Let(V,F,+,·)is a linear space,definition of the generalized power set operator is given in P*(V)(?)P(V)-Φ:
A(?)B=C∈P*(V); definition of the generalized power set operator is given in P×P*(V):
k(?)A=D∈P*(V), where k∈F,A,B∈P*(V).
Definitions 3.2.1 Let(V,F,+,·)is a linear space,definition of the generalized power set operator is given in P*(V)(?)P(V)-Φ:
A(?)B=C∈P*(V); definition of the generalized power set operator is given in P×P*(V):
k(?)A=D∈P*(V), where k∈F,A,B∈P*(V).If(P*(X),F,(?),(?))_1 is linear space with two operations, then we call that it is a first generalized power linear space in number field F.
Definitions 3.2.2 Let X is a non- Empty Set,F is a number field.Definition of the generalized power set operator is given in P*(X)(?)P(X):
A(?)B∈P*(X); definition of the generalized power set operator is given in P×P*(V):
k(?)A∈P*(X), where k∈F,A,B∈P*(V).If(P*(X),F,(?),(?))_2 is linear space with two operations,then we call that it is a second generalized power linear space in number field F.In the general,it is also known as power-set linear space.
The first generalized power linear space and the second generalized power linear space is commonly known as generalized power linear space.
Correspondingly,Some concepts are defined in generalized power linear space: generalized power linear presentation,generalized power linear dependent,generalized power linear independent,power radix,dimension,generalized power subspace, homomorphism of generalized power linear space,isomorphism of generalized power linear space and their serial nature.
We discussed power set upgrade and generalized power set upgrade of operators, before we have obtained a series of super-structure,and nature of super-structure and a series of results,such as.the power group,the power ring,the power latice,power mode and power linear space.Despite the series of super-structure is same with the corresponding,original structure,but widening the corresponding content and applications.
首先线性空间上的线性运算的幂集提升
设(V,F,+,·)是一个线性空间,在P*(V)(?)P(V)-Φ中定义二元集运算(称之为二元幂加法):
A(?)B={a+b|a∈A,b∈B};二元幂数量乘积运算(称之为二元幂数量乘积):
k(?)A={ka|a∈A},其中k∈F,A,B∈P*(V)。
定义2.3.1设(V,F,+,·)是一个线性空间,在P*(V)(?)P(V)-Φ中定义二元幂加法
A(?)B={a+b|a∈A,b∈B};二元幂数量乘积运算:
k(?)A={ka|a∈A},其中k∈F,A,B∈P*(V),如果P*(V)关于以上两个运算构成数域F上的线性空间,那么称P*(V)是由数域F上线性空间V诱导出的一个自然幂线性空间。
定理2.3.1自然幂线性空间一定是原线性空间或其某个子空间。
定义2.3.2设(V,F,+,·)是一个线性空间,在P*(V)(?)P(V)-Φ中定义二元幂加法
A(?)B={a+b|a∈A,b∈B};二元幂数量乘积运算:
k(?)A={ka|a∈A}(k≠0,特别地当k=0时,规定0(?)A=0),其中k∈F,A,B∈P*(V),如果P*(V)关于以上两个运算构成数域F上的线性空间,那么称P*(V)是由数域F上线性空间V诱导出的一个幂线性空间。
自然幂线性空间和幂线性空间统称为幂线性空间。
自然幂线性空间必是幂线性空间的特例。
定理2.3.2幂线性空间P*(V)一定是原线性空间V或其子空间的一个商空间。
定义2.4.1设(P*(V),F,(?),(?))是由线性空间(V,F,+,·)诱导出的一个幂线性空间,且A,A_1,A_2,…,A_s∈P*(V),若(?)k_1,k_2,…,k_s∈F,使得:
k_1A_1+k_2A_2+…+k_sA_s=A则称A可由幂向量组A_1,A_2,…,A_s幂线性表出,k_1A_1+k_2A_2+…+k_sA_s称为幂向量组A_1,A_2,…,A_s幂线性组合。
定义2.4.2设(P*(V),F,(?),(?))是由数域F上n维线性空间(V,F,+,·)诱导出的一个幂线性空间,且A_1,A_2,…,A_s∈P*(V),若(?)k_1,k_2,…,k_s∈F不全为零,使得:
k_1A_1+k_2A_2+…+k_sA_s=0则称幂向量组A_1,A_2,…,A_s是幂线性相关的;否则称为幂线性无关。
定义2.4.3设(P*(V),F,(?),(?))是由数域F上n维线性空间(V,F,+,·)诱导出的一个幂线性空间,且A_1,A_2,…,A_m∈P*(V),若满足:
(1)A_1,A_2,…,A_m是幂线性无关的;
(2)(?)A∈P*(V)都可由幂向量组A_1,A_2,…,A_m幂线性表出则称幂向量组A_1,A_2,…,A_m是幂线性空间(P*(V),F,(?),(?))的一组幂基,简称为基,并称幂线性空间(P*(V),F,(?),(?))是m维的,记为:dim(P*(V))=m。
定理2.4.1幂线性空间的维数等于线性空间或其子空间的维数与幂零元的维数之差。
定理2.4.2自然幂线性空间的维数等于线性空间或其子空间的维数。
定义2.5.1设(P*(V),F,(?),(?))数域F上n维线性空间V的幂线性空间,W为P*(V)的非空子集,若(W,F,(?),(?))构成数域F上的幂线性空间,则称W为P*(V)的幂线性子空间,简称幂子空间。
定义2.5.2设W_1,W_2是(P*(V),F,(?),(?))的幂线性子空间,则称W_1∩W_2为P*(V)的幂子空间W_1与W_2的幂交。
定义2.5.3设W_1,W_2是(P*(V),F,(?),(?))的幂线性子空间,则称W_1+W_2为P*(V)的幂子空间W_1与W_2的幂和。
定理2.5.1设W_1,W_2是(P*(V),F,(?),(?))的幂子线性空间,则W_1∩W_2也是P*(V)的幂子空间,即:幂子空间的的幂交仍为幂子空间。
定理2.5.2设W_1,W_2是(P*(V),F,(?),(?))的幂子线性空间,则W_1+W_2也是P*(V)的幂子空间,即幂子空间的的幂和仍为幂子空间。
定理2.5.3设W_1,W_2是(P*(V),F,(?),(?))的幂子线性空间,则
dim(W_1)+dim(W_2)=dim(W_1∩W_2)+dim(W_1+W_2)。(此为幂线性空间的维数公式。)
定义2.6.1设(P*_1(V),F,(?)_1,(?)_1)与(P*_2(W),F,(?)_2,(?)_2)分别是线性空间(V,F,+_1,·_1)与(W,F,+_2,·_2)的幂线性空间,如果P*_1(V)到P*_2(W)的一个线性映射σ满足:
(1)σ是满射;
(2)σ(A_1(?)_1A_2)=σ(A_1)(?)_2σ(A_2);
(3)σ(k(?)_1A_1)=k(?)_2σ(A_1)其中A_1,A_2∈P*_1(V),k∈F,则σ称是P*_1(V)到P*_2(W)的同态映射,也称P*_1(V)与P*_2(W)是同态的。
定义2.6.2设(P*_1(V),F,(?)_1,(?)_1)与(P*_2(W),F,(?)_2,(?)_2)分别是线性空间(V,F,+_1,·_1)与(W,F,+_2,·_2)的幂线性空间,如果P*_1(V)到P*_2(W)的一个线性映射σ满足:
(1)σ是双射;
(2)σ(A_1(?)_1A_2)=σ(A_1)(?)_2σ(A_2);
(3)σ(k(?)_1A_1)=k(?)_2σ(A_1)其中A_1,A_2∈P*_1(V),k∈F,则σ称是P*_1(V)到P*_2(W)的同构映射,也称P*_1(V)与P*_2(W)是同构的。
定理2.6.1同一个数域下的两个有限维的幂线性空间同构当且仅当它们维数是相同的。
定理2.6.2任意线性空间与它诱导出的幂线性空间同态。
定理2.6.3任意幂线性空间都与其原线性空间的某个子空间同构。
推论同一个数域下的任意两个线性空间或者幂线性空间的维数相等,则它们必同构。
对线性空间的线性运算的广义幂集提升,一般集合上定义幂集线性运算,设(V,F,+,·)是一个线性空间,在P*(V)(?)P(V)-Φ中定义二元集运算(称之为二元幂加法):
A(?)B=C∈P*(V);二元幂数集乘积运算:
k(?)A=D∈P*(V),其中k∈F,A,B∈P*(V)。
定义3.2.1设(V,F,+,·)是一个线性空间,在P*(V)(?)P(V)-Φ中定义二元幂加法和幂数量乘积运算,使得:A(?)B=C∈P*(V),k(?)A=D∈P*(V),其中k∈F,A,B∈P*(V),如果P*(V)关于以上两个运算构成数域F上的线性空间,则称P*(V)为数域F上线性空间V的第一广义幂线性空间,简称V上的第一广义幂线性空间,记为:(P*(V),F,(?),(?))_1。
定义3.2.2设X是非空集合,F是数域,在P*(X)(?)P(X)-Φ与F中定义二元集加法和幂数集乘积运算:A(?)B∈P*(X),k(?)A∈P*(X),其中k∈F,A,B∈P*(X),若P*(X)关于以上两个运算构成数域F上的线性空间,则称P*(X)为数域F上线性空间V的第二广义幂线性空间,简称V上的第二广义幂线性空间,一般也称为幂集线性空间,记为:(P*(X),F,(?),(?))_2。
第一广义幂线性空间与第二广义幂线性空间通称为广义幂线性空间。为方便起见,统一记广义幂线性空间为(P*(X),F,(?),(?))。
相应的可以定义广义幂线性空间上的幂线性表出、幂线性相关、幂线性无关、幂基、维数、广义幂子空间、同态与同构以及系列性质。
运算的提升可以得出各种超结构,如幂群、幂环、幂格、幂模等,当然也包括幂线性空间。尽管提升后的超结构一般来讲并未脱离原有结构的范畴,但却拓宽了相应的内涵和应用范围。
With the development of fuzzy math,the importance of set-valued function is growing,it need to upgrade that Various mathematical structures from the universe to power set.Since Li Hongxing professor in the literature[1]consider the upgrade of the algebraic structure,and the concept of the HX- group is given at first,upgrading of algebraic group- power group is discussed in literature[2].More and more scholars and enthusiasts concern studying of super algebra,A series of very good results of power group is given in the literature[3]-[23],The power set upgrade of the ring-power ring is discussed in the literature[24]-[38],and The power set upgrade of lattic-power lattic is discussed in the literature[39]-[43],and a series of valuable results is given,the usual methods of studying super-algebra structure:(1)algebra operators upgrade;(2) the concept and examples of power algebraic structure;(3)the nature of power algebraic structure.
First,in this paper,power set upgrade of linear operators is discussed in linear space.Let(V,F,+,·)is a linear space,definition of the power set operator is Given in P*(V)(?)P(V)-Φ:
A(?)B={a+b|a∈A,b∈B}; definition of the power set operator is Given in P×P*(V):
k(?)A={ka|a∈A}, where k∈F,A,B∈P*(V).
Definitions 2.3.1 Let(V,F,+,·)is a linear space,definition of the power set operator is given in P*(V)(?)P(V)-Φ:
A(?)B={a+b|a∈A,b∈B}; definition of the power set operator is Given in P×P*(V):
k(?)A={ka|a∈A}, where k∈F,A,B∈P*(V).If(P*(V),F,(?),(?))is linear space with two operations, then We call that it is a natural power linear space in number field F.
Theorem 2.3.1 natural power linear space is the original linear space or a subspace of the original linear space.
Definitions 2.3.2 Let(V,F,+,·)is a linear space,definition of the power set operator is Given in P*(V)(?)P(V)-Φ:
A(?)B={a+b|a∈A,b∈B}; definition of the power set operator is Given in P×P*(V):
k(?)A={ka|a∈A}, where k(≠0)∈F(Especially,if k=0,0(?)A=0),A,B∈P*(V).If(P*(V),F,(?),(?))is linear space with two operations,then We call that it is a power linear space in number field F.
Natural power linear space and power linear space space power collectively referred to as power linear space.
It is a special case that natural power linear space is power linear space.
Theorem 2.3.2 power linear space is a quotient space of the original linear space or it's sub-space.
Definitions 2.4.1 Let(P*(V),F,(?),(?))is a power linear space,and
A,A_1,A_2,...,A_s∈P*(V), If(?)k_1,k_2,...,k_s∈F,such that:
k_1A_1+k_2A_2+...+k_sA_s=A, then A is claimed power-linear presentation by the power vector group A_1,A_2,...,A_s, In this case,k_1A_1+k_2A_2+...+k_sA_2≤is claimed power linear combination of power vector group A_1,A_2,...,A_s.
Definition 2.4.2 Let(P*(V),F,(?),(?))is a power linear space,and
A_1,A_2,...,A_s∈P*(V), If(?)k_1,k_2,...,k_s∈F,and k_1≠0 or k_2≠0 or...k_s≠0,such that:
k_1A_1+k_2A_2+...+k_sA_s=0, then the power vector group A_1,A_2,...,A_s is claimed power linear dependent, otherwise,known as the power linearly independent.
Definition 2.4.3 Let(P*(V),F,(?),(?))is a power linear space,and
A_1,A_2,...,A_m∈P*(V), and satisfy that:
(1)A_1,A_2,...,A_s is power linearly independent;
(2)(?)A∈P*(V),all is the power linear combination of A_1,A_2,...,A_s, then A_1,A_2,...,A_s is claimed a power base in(P*(V),F,(?),(?)).In this base,saying power linear space(P*(V),F,(?),(?))is m-dimensional linear space.
Theorem 2.4.1 The dimension of power linear space equivalent to the poor of the dimension of the original linear space and the dimension of O,or the poor of the dimension of sub-space of the original linear space and the dimension of O.
Theorem 2.4.2 The dimension of natural power linear space equivalent to the dimension of the original linear space or dimension of sub-space of the original linear space.
Definitions 2.5.1 Let(P*(V),F,(?),(?))is a power linear space,and W(?)P*(V),If W is a power linear space,then We call that W is a power linear sub-space of P*(V).
Definitions 2.5.2 Let W_1,W_2 is a power linear sub-space of(P*(V),F,(?),(?)), then We call that W_1∩W_2 is the power intersection of power sub-space W_1 and W_2.
Theorem 2.5.1 Let W_1,W_2 is a power linear sub-space of(P*(V),F,(?),(?)),then it is power linear sub-space that the power intersection of W_1 and W_2.
Theorem 2.5.3 Let W_1,W_2 is a power linear sub-space of(P*(V),F,(?),(?)),then
dim(W_1)+dim(W_2)=dim(W_1∩W_2)+dim(W_1+W_2)。
Definitions 2.6.1 Let(P*_1(V),F,(?)_1,(?)_1)and(P*2(W),F,(?)_2,(?)_2)are two power linear space,σis a linear map for P*_1(V)to P*_2(W),if satisfy:
(1)σIs Surjectivity;
(2)σ(A_1,(?)_1,A_2)=σ(A_1)(?)_2σ(A_2);
(3)σ(k(?)_1A_1)=k(?)_2σ(A_1), where A_1,A_2∈P*_1(V),k∈F,then,σis claimed to homomorphism mapping for P*_1(V)to P*_2(W).
Definitions 2.6.2 Let(P*_1(V),F,(?)_1,(?)_1)and(P*_2(W),F,(?)_2,(?)_2)are two power linear space,σis a linear map for P*_1(V)to P*_2(W),if satisfy:
(1)σBijective;
(2)σ(A_1(?)_1A_2)=σ(A_1)(?)_2σ(A_2);
(3)σ(k(?)_1A_1)=k(?)_2σ(A_1) where A_1,A_2∈P*_1(V),k∈F,then,σis claimed to isomorphism mapping for P*_1(V)to P*_2(W).
Theorem 2.6.1 Under same number field,the two finite dimensional power linear space is isomorphism if and only if their dimension is the same.
Then,generalized power set upgrade of linear operators is discussed in linear space. Let(V,F,+,·)is a linear space,definition of the generalized power set operator is given in P*(V)(?)P(V)-Φ:
A(?)B=C∈P*(V); definition of the generalized power set operator is given in P×P*(V):
k(?)A=D∈P*(V), where k∈F,A,B∈P*(V).
Definitions 3.2.1 Let(V,F,+,·)is a linear space,definition of the generalized power set operator is given in P*(V)(?)P(V)-Φ:
A(?)B=C∈P*(V); definition of the generalized power set operator is given in P×P*(V):
k(?)A=D∈P*(V), where k∈F,A,B∈P*(V).If(P*(X),F,(?),(?))_1 is linear space with two operations, then we call that it is a first generalized power linear space in number field F.
Definitions 3.2.2 Let X is a non- Empty Set,F is a number field.Definition of the generalized power set operator is given in P*(X)(?)P(X):
A(?)B∈P*(X); definition of the generalized power set operator is given in P×P*(V):
k(?)A∈P*(X), where k∈F,A,B∈P*(V).If(P*(X),F,(?),(?))_2 is linear space with two operations,then we call that it is a second generalized power linear space in number field F.In the general,it is also known as power-set linear space.
The first generalized power linear space and the second generalized power linear space is commonly known as generalized power linear space.
Correspondingly,Some concepts are defined in generalized power linear space: generalized power linear presentation,generalized power linear dependent,generalized power linear independent,power radix,dimension,generalized power subspace, homomorphism of generalized power linear space,isomorphism of generalized power linear space and their serial nature.
We discussed power set upgrade and generalized power set upgrade of operators, before we have obtained a series of super-structure,and nature of super-structure and a series of results,such as.the power group,the power ring,the power latice,power mode and power linear space.Despite the series of super-structure is same with the corresponding,original structure,but widening the corresponding content and applications.
引文
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[9]林楠,幂群与拟商群[J]辽宁师范大学学报(自然科学版),1998,(02)
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[11]Zhang Zhenliang.Classification of HX groups and their ehains of normal subgroups[J].Italian Journal of pure and Applied Mathematics,1999(2),125-134.
[12]张振良 幂群的结构与分类[J]纯粹数学与应用数学 1998(1)
[13]彭家寅 幂等Fuzzy半群与拟Fuzzy商群[J]内江师范学院学报 2004(04)
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[19]Zhenliang Zhang.Classifications of HX Groups and Their Chair of Normal Subgroup.Rivista ai Matematica pura ed applicata(意刊),1998,23
[20]李扉,张振良 点态化的模糊幂群[C]第12届全国模糊系统与模糊数学学术年会论文集,2004
[21]张诚一,党平安Fuzzy幂群的基数及表示[J]模糊系统与数学,2001,(04)
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[24]李洪兴 HX环 数学季刊,1991
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