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Increasingly prominent with the development of fuzzy mathematics, set the value of the importance of mapping the the various mathematical structure has a need to enhance its power set by the domain. few scholars and enthusiasts attention [3] - [23] to get the a series powergroups good results [24] - [38] gives the ring of power set to enhance - power ring and get The power ring, a series of nature, literature [39] - [43] gives the power set to enhance grid - the power grid, and a corresponding series of valuable results. Algebraic structure of the usual research methods: (1) the algebraic operations improved; (2) the concept of power algebra structure instance; (3) the nature of the power algebra structure. The first linear operation of the power set on the linear space enhance the set (V, F,, ·) is a linear space, Binary set operations defined in the P * (V) (?) P (V)-Φ ( called binary power adder): A (?) B = {ab | a ∈ A, b ∈ B}; binary power of the number of product operations (called binary power of the number of the product): k (?) A = {ka | a ∈ A}, where k ∈ F, A, B ∈ P * (V). Defined 2.3.1 disposed (V, F,, ·) is a linear space and in the definition of P * (V) (?) P (V)-Φ in the binary exponentiation adder A (?) B = {AB | A ∈ A, b ∈ B}; the binary power quantities multiplication operator: k (?) A = {KA | a ∈ A}, where k ∈ F, A, B ∈ P * (V), P * (V) linear space on the above two operations constitute a number field F, then known as P * (V) is a natural power induced by a number field F on the linear space V linear space. Theorem 2.3.1 natural power linear space must be the original linear space or a subspace. Defined 2.3.2 disposed (V, F,, ·) is a linear space and in the definition of P * (V) (?) P (V)-Φ in the binary exponentiation adder A (?) B = {AB | A ∈ A, b ∈ B}; the binary power quantities multiplication operator: k (?) A = {KA | a ∈ A} (k ≠ 0, particularly when k = 0, require 0 (?), A = 0) k ∈ F, A, B ∈ P * (V), P * (V) linear space on the above two operations constitute a number field F, then known as P * (V) is the line number field F Space V induced a power linear space. The natural power linear space and power linear space collectively referred to as the power of linear space. The natural power linear space must be a special case of the power of linear space. Theorem 2.3.2 power linear space P * (V) must be a quotient space of the original linear space V or subspace. The definition 2.4.1 set (P * (V), F (?), (?)) Is induced by a linear space (V, F,, ·) out of a power-linear space, and A, of A 1 , A 2 , ..., A s ∈ P * (V), if (?) k 1 k 2 , ..., k s ∈ F, such that: k 1 A 1 k 2 A 2 ... k s A the s = A then A may be the power of vector group A 1 , A power linear Expressed 2 , ..., A s , k 1 A 1 k 2 < / sub> A 2 ... k s the A s is called the power of Vector Group A 1 A 2 , ..., A s the power linear combination. Definition 2.4.2 (P * (V), F (?), (?)) By the number field F on a power of n-dimensional linear space (V, F,, ·) induced linear space, and A 1 , A 2 , ... the A s ∈ P * (V), the if (?) k the 1 k 2 , ..., k s ∈ F not all zero, such that: k , A , 1 1 k 2 A 2 ... k the s A s = 0 claimed power of with Vector Group A 1 < / sub>, A 2 , ..., A s is the power linearly related; otherwise known as power linear. Definition 2.4.3 (P * (V), F (?), (?)) By the number field F on a power of n-dimensional linear space (V, F,, ·) induced linear space, and A 1 , A 2 , ..., A m ∈ P * (V), if the meet: (1) A 1 , A 2 , ..., A m is the power linearly independent; (2) (?) A ∈ P * (V) can be by the power of vector A 2 in the Group A 1, ..., the A m power linear Expressed claimed power of with Vector Group A 1 , A 2 , ..., A m the power linear space (P * (V), F (?), (?)) the power of a group of group , referred to as the group, and said power linear space (P * (V), F, (?), (?)) is the m-dimensional, denoted as: Dim (P * (v)) = m. Of Theorem 2.4.1 power linear space of dimension equal to the dimension of the linear space or subspace nilpotent $ dimension of difference. Theorem 2.4.2 natural power linear space dimension equal to the dimension of the linear space or subspace. Definition 2.5.1 (P * (V), F (?), (?)) Number field F on n-dimensional linear space V power linear space, W is a nonempty subset of P * (V) (W, F, (?), (?)) the power linear space constitute a number field F, then W is P * (V) of the exponentiation linear subspace, referred to as the power sub-space. Definition 2.5.2 W 1 W 2 is (P * (V), F, (?), (?)) A power of linear subspace, then W 1 ∩ W 2 P * (V) power of subspace W and W 2 a power cross. Definition 2.5.3 W 1 W 2 is (P * (V), F, (?), (?)) A power of linear subspace, then W 1 W 2 and W 2 P * (V) power subspace W power and . Theorem 2.5.1 Let W 1 W 2 (P * (V), F (?) (?)) Power sub-linear space, W 1 ∩ W 2 P * (V) of the power sub-space, namely: power Subspace the power cross remains exponentiation subspace. Theorem 2.5.2 W 1 W 2 (P * (V), F, (?), (?)) Power sub-linear space, W 1 W 2 is also the P * (V) of the power sub-space, i.e. the power of the sub-space of a power and remains power sub-space. Theorem 2.5.3 Let W 1 W 2 (P * (V), F (?), (?)) Power sub-linear space, dim (W 1 ) dim (W 2 ) = dim (W 1 ∩ W 2 ) dim (W 1 W 2 ). (This is the power of the linear space of dimension formula.) Definition 2.6.1 (the P * 1 (V), F (?) 1 (?) 1 ) and (P * 2 (W), F (?) the 2 (?) 2 ) are linear space (V, F, 1 the · 1 ) and (W, F, 2 , the · 2 ) a power of linear space, If P * is 1 (V) to P * 2 (W) of a linear mapping σ met: (1) The σ Surjectivity; (2) σ (A 1 (?) 1 A 2 ) = σ (A 1 ) (?) 2 σ (A 2 ); (3) σ (k (?) 1 A 1 < / sub>) = k (?) 2 σ (A 1 ) A 1 A 2 ∈ P of * 1 (V), k ∈ F, then σ P * 1 (V) to P * 2 (W) of the Homomorphisms, also known as the P * 1 (V) with P * 2 (W) is the same state. Definition 2.6.2 (P of * 1 (V), F (?) 1 (?) 1 ) and (P * 2 (W), F, (?) to 2 , (?) 2 ) is a linear space (V, F, respectively, 1 , · 1 ) and (W, F, 2 , · , 2 ) power linear space, if P * 1 (V) to the P * 2 (W) of a linear mapping σ satisfies: (1) σ is bijective; (2) σ (A 1 (?) 1 A 2 ) = σ (A 1 ) (?) 2 σ (A 2 ); (3) σ (k (?) 1 A 1 ) = k (?) 2 σ (A 1 ) wherein A a 1 , A 2 ∈ P * 1 (V) , k ∈ F, then σ P of * 1 (V) to P * 2 (W) of the isomorphic mapping, also known P * 1 < / sub> (V) with P * 2, (W) is isomorphic. Theorem 2.6.1 the same number of domains under a power of two finite-dimensional linear space isomorphic if and only if they dimension is the same. Theorem 2.6.2 any linear space with induced the power linear space homomorphism. Theorem 2.6.3 any power linear space with its original linear space a subspace isomorphic. Infer any two linear space under the domain of the same number or power of linear space of dimension equal, they must be isomorphic. Generalized power set to enhance the linear operation of the linear space, generally on the collection define power set of linear operation, the set (V, F,, ·) is a linear space, the P * (V) (?) P ( V)-Φ defined in the dual set computing (called two yuan power adder): A (?) B = C ∈ P * (V); the binary exponent Set product operation: k (?) A = D ∈ P * (V), where k ∈ F, A, B ∈ P * (V). Definition 3.2.1 set (V, F,, ·) is a linear space, the multiplication operation is defined in the P * (V) (?) P (V)-Φ the binary exponentiation adder and power quantity, so that: to a ( ?) B = C ∈ P * (V), k (?) A = D ∈ P * (V), where k ∈ F, A, B ∈ P * (V), if P * (V), on the above two op constitute a linear space number field F, then P * (V) a field F of linear space V number generalized the power linear space, referred to as V on the first generalized power linear space, denoted by: (P * (V), F, (?), (?)) 1 . X is a non-empty set defined 3.2.2, F is the number of domains, in the P * (X) (?) P (X)-φ with F defined binary set addition and exponent set product computation: A (?) ∈ P * (X), k (?) A ∈ P * (X), where k ∈ F, A, B ∈ P * (X), P * (X) on the above two expressions constitute number field F linear space, then amounting field F of P * (X) on the linear space V Generalized power linear space, referred to as V on the second generalized power linear space, generally called the power set of linear space, referred to as: (P * (X), F, (?), (?)) 2 . The first Generalized power linear space with the second generalized the power linear space known as the Generalized power linear space. For convenience, the unified Hutchison Generalized power linear space (P * (X), F (?), (?)). Can define the power of linear Expressed on the broad powers of linear space, power linear power linearly independent, the nature of the power basis, dimension, generalized power sub-space, homomorphism and isomorphism series. Computing to enhance a variety of super-structure, such as a power group, power ring, power grid, power mode, etc. can be drawn, of course, including the power of linear space. Enhanced super-structure is generally speaking not out of the scope of the original structure, but to broaden the the appropriate connotation and range of applications.
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