| 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,andA,A1,A2,...,As∈P*(V), If(?)k1,k2,...,ks∈F,such that:k1A1+k2A2+...+ksAs=A, then A is claimed power-linear presentation by the power vector group A1,A2,...,As, In this case,k1A1+k2A2+...+ksA2≤is claimed power linear combination of power vector group A1,A2,...,As.Definition 2.4.2 Let(P*(V),F,(?),(?))is a power linear space,andA1,A2,...,As∈P*(V), If(?)k1,k2,...,ks∈F,and k1≠0 or k2≠0 or...ks≠0,such that:k1A1+k2A2+...+ksAs=0, then the power vector group A1,A2,...,As 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,andA1,A2,...,Am∈P*(V), and satisfy that:(1)A1,A2,...,As is power linearly independent;(2)(?)A∈P*(V),all is the power linear combination of A1,A2,...,As, then A1,A2,...,As 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 W1,W2 is a power linear sub-space of(P*(V),F,(?),(?)), then We call that W1∩W2 is the power intersection of power sub-space W1 and W2.Theorem 2.5.1 Let W1,W2 is a power linear sub-space of(P*(V),F,(?),(?)),then it is power linear sub-space that the power intersection of W1 and W2.Theorem 2.5.3 Let W1,W2 is a power linear sub-space of(P*(V),F,(?),(?)),thendim(W1)+dim(W2)=dim(W1∩W2)+dim(W1+W2)。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(A1,(?)1,A2)=σ(A1)(?)2σ(A2);(3)σ(k(?)1A1)=k(?)2σ(A1), where A1,A2∈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(A1(?)1A2)=σ(A1)(?)2σ(A2);(3)σ(k(?)1A1)=k(?)2σ(A1) where A1,A2∈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.
|
PDF Full Text Request