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k Definition 1.2 Let V = Vi be a complete multi-space with set e iS=1 k O(V )= {(+˙ i, ·i) | 1 ≤ i ≤ m} and F = Fi a multi-filed space with double binary e e iS=1 operation set O(F ) = {(+i, ×i) | 1 ≤ i ≤ k}. If for any integers i, j, 1 ≤ i, j ≤ k e and ∀a, b, c ∈ V , k1,k2 ∈ F , e e (i) (Vi; +˙ i, ·i) is a vector space on Fi with vector additive +˙ i and scalar multipli- cation ·i; (ii) (a+˙ ib)+˙ jc = a+˙ i(b+˙ jc); (iii) (k1 +i k2) ·j a = k1 +i (k2 ·j a); if all those operation results exist, then V is called a multi-vector space on the multi- filed space F with a binary operation sete O(V ), denoted by (V ; F ). e e e e For V1 ⊂ V and F1 ⊂ F ,if(V1; F1) is also a multi-vector space, then call e e e e e e (V1; F1) a multi-vector subspace of (V ; F ). e Thee subject of this paper is to finde somee characteristics of a multi-vector space. For terminology and notation not defined here can be seen in [1], [3] for linear alge- braic terminologies and in [2], [4] − [11] for multi-spaces and logics.

2. Characteristics of a multi-vector space

First, we have the following result for multi-vector subspace of a multi-vector space. Theorem 2.1 For a multi-vector space (V ; F ), V1 ⊂ V and F1 ⊂ F , (V1; F1) is a multi-vector subspace of (V ; F ) if and onlye e if fore anye vectore additivee +e˙ , scalare e e multiplication · in (V1; F1) and ∀a, b ∈ V , ∀α ∈ F , e e e e

α · a+˙ b ∈ V1 e if their operation result exist.

k k k Proof Denote by V = Vi, F = Fi. Notice that V1 = (V1 Vi). By e iS=1 e iS=1 e iS=1 e T definition, we know that (V1; F1) is a multi-vector subspace of (V ; F ) if and only if e e e e for any integer i, 1 ≤ i ≤ k, (V1 Vi; +˙ i, ·i) is a vector subspace of (Vi, +˙ i, ·i) and F1 e T e is a multi-filed subspace of F or V1 Vi = ∅. e e T

2 According to the criterion for linear subspaces of a linear space ([3]), we know that for any integer i, 1 ≤ i ≤ k, (V1 Vi; +˙ i, ·i) is a vector subspace of (Vi, +˙ i, ·i) if e T and only if for ∀a, b ∈ V1 Vi, α ∈ Fi, e T α a ˙ b V V . ·i +i ∈ 1 \ i e That is, for any vector additive +,˙ · in (V1; F1) and ∀a, b ∈ V , e e e ∀α ∈ F , if α · a+˙ b exists, then α · a+˙ b ∈ V1. ♮ e e Corollary 2.1 Let (U; F1), (W ; F2) be two multi-vector subspaces of a multi-vector e e f e space (V ; F ). Then (U W ; F1 F2) is a multi-vector space. e e e T f e T e For a multi-vector space (V ; F ), vectors a1, a2, ··· , an ∈ V , if there are scalars e e e α1, α2, ··· , αn ∈ F such that e

α1 ·1 a1+˙ 1α2 ·2 a2+˙ 2 ··· +˙ n−1αn ·n an = 0, where 0 ∈ V is an unit under an operation + in V and +˙ i, ·i ∈ O(V ), then the e e e vectors a1, a2, ··· , an are said to be linearly dependent. Otherwise, a1, a2, ··· , an to be linearly independent. Notice that in a multi-vector space, there are two cases for linearly independent vectors a1, a2, ··· , an:

(i) for any scalars α1, α2, ··· , αn ∈ F , if e

α1 ·1 a1+˙ 1α2 ·2 a2+˙ 2 ··· +˙ n−1αn ·n an = 0, where 0 is a unit of V under an operation + in O(V ), then α1 = 0+1 , α2 = e e 0+2 , ··· , αn = 0+n , where 0+i , 1 ≤ i ≤ n are the units under the operation +i in F . e (ii) the operation result of α1 ·1 a1+˙ 1α2 ·2 a2+˙ 2 ··· +˙ n−1αn ·n an does not exist.

Now for a S ⊂ V , define its linearly spanning set DSE to be b e b

DSE = { a | a = α1 ·1 a1+˙ 1α2 ·2 a2+˙ 2 ···∈ V , ai ∈ S, αi ∈ F, i ≥ 1}. b e b e

For a multi-vector space (V ; F ), if there exists a subset S, S ⊂ V such that V = DSE, e e b b e e b then we say S is a linearly spanning set of the multi-vector space V . If the vectors in a linearlyb spanning set S of the multi-vector space V are linearlye independent, then S is said to be a basisbof V . e b e Theorem 2.2 Any multi-vector space (V ; F ) has a . e e k k Proof Assume V = Vi, F = Fi and the basis of the vector space (Vi; +˙ i, ·i) e iS=1 e iS=1 is ∆i = {ai1, ai2, ··· , aini }, 1 ≤ i ≤ k. Define

3 k . ∆ = [ ∆i b i=1 Then ∆ is a linearly spanning set for V by definition. If vectorsb in ∆ are linearly independent,e then ∆ is a basis of V . Otherwise, b b e choose a vector b1 ∈ ∆ and define ∆1 = ∆ \{b1}. b b b If we have obtained the set ∆s,s ≥ 1 and it is not a basis, choose a vector b bs+1 ∈ ∆s and define ∆s+1 = ∆s \{bs+1}. b b b If the vectors in ∆s+1 are linearly independent, then ∆s+1 is a basis of V . Other- b b e wise, we can define the set ∆s+2. Continue this process. Notice that for any integer b i, 1 ≤ i ≤ k, the vectors in ∆i are linearly independent. Therefore, we can finally get a basis of V . ♮ Now we considere the finite-dimensional multi-vector space. A multi-vector space V is finite-dimensional if it has a finite basis. By Theorem 2.2, if for any integer e i, 1 ≤ i ≤ k, the vector space (Vi;+i, ·i) is finite-dimensional, then (V ; F ) is finite- e e dimensional. On the other hand, if there is an integer i0, 1 ≤ i0 ≤ k, such that the vector space (Vi0 ;+i0 , ·i0 ) is infinite-dimensional, then (V ; F ) is infinite-dimensional. This enables us to get the following corollary. e e

k k Corollary 2.2 Let (V ; F ) be a multi-vector space with V = Vi, F = Fi. Then e e e iS=1 e iS=1 (V ; F ) is finite-dimensional if and only if for any integer i, 1 ≤ i ≤ k, (Vi;+i, ·i) is finite-dimensional.e e

Theorem 2.3 For a finite-dimensional multi-vector space (V ; F ), any two bases have the same of vectors. e e

k k Proof Let V = Vi and F = Fi. The proof is by the induction on k. For e iS=1 e iS=1 k = 1, the assertion is true by Theorem 4 of Chapter 2 in [3]. For the case of k = 2, notice that by a result in linearly vector space theory (see also [3]), for two subspaces W1, W2 of a finite-dimensional vector space, if the basis of W W is {a , a , ··· , at}, then the basis of W W is 1 T 2 1 2 1 S 2

{a1, a2, ··· , at, bt+1, bt+2, ··· , bdimW1 , ct+1, ct+2, ··· , cdimW2 }, where, {a1, a2, ··· , at, bt+1, bt+2, ··· , bdimW1 } is a basis of W1 and {a1, a2, ··· , at, ct+1, ct+2, ··· , cdimW2 } a basis of W2. Whence, if V = W1 W2 and F = F1 F2, then the basis of V is also e S e S e

{a1, a2, ··· , at, bt+1, bt+2, ··· , bdimW1 , ct+1, ct+2, ··· , cdimW2 }. Assume the assertion is true for k = l, l ≥ 2. Now we consider the case of k = l + 1. In this case, since

4 l l V V V , F F F , =( [ i) [ l+1 =( [ i) [ l+1 e i=1 e i=1 by the induction assumption, we know that any two bases of the multi-vector space l l l ( Vi; Fi) have the same number p of vectors. If the basis of ( Vi) Vl+1 is iS=1 iS=1 iS=1 T {e1, e2, ··· , en}, then the basis of V is e

{e1, e2, ··· , en, fn+1, fn+2, ··· , fp, gn+1, gn+2, ··· , gdimVl+1 }, l l where {e1, e2, ··· , en, fn+1, fn+2, ··· , fp} is a basis of ( Vi; Fi) and {e1, e2, ··· , en, iS=1 iS=1 gn+1, gn+2, ··· , gdimVl+1 } a basis of Vl+1. Whence, the number of vectors in a basis of V is p + dimVl+1 − n for the case n = l + 1. eTherefore, by the induction principle, we know the assertion is true for any integer k. ♮ The number of a finite-dimensional multi-vector space V is called its , denoted by dimV . e e k Theorem 2.4(dimensional formula) For a multi-vector space (V ; F ) with V = Vi e e e iS=1 k and F = Fi, the dimension dimV of V is e iS=1 e e k dimV i−1 dim V V V . = X(−1) X ( i1 \ i2 \ ··· \ ii) e i=1 {i1,i2,···,ii}⊂{1,2,···,k}

Proof The proof is by induction on k. For k = 1, the formula is the trivial case of dimV = dimV1. for k = 2, the formula is e dimV dimV dimV dim V dimV , = 1 + 2 − ( 1 \ 2) e which is true by Theorem 6 of Chapter 2 in [3]. Now assume the formula is true for k = n. Consider the case of k = n + 1. According to the proof of Theorem 2.15, we know that

n n dimV dim V dimV dim V V = ( [ i)+ n+1 − (( [ i) \ n+1) e i=1 i=1 n n dim V dimV dim V V = ( [ i)+ n+1 − ( [ ( i \ n+1)) i=1 i=1 n dimV i−1 dim V V V = n+1 + X(−1) X ( i1 \ i2 \ ··· \ ii) i=1 {i1,i2,···,ii}⊂{1,2,···,n}

5 n i−1 dim V V V V + X(−1) X ( i1 \ i2 \ ··· \ ii \ n+1) i=1 {i1,i2,···,ii}⊂{1,2,···,n} n i−1 dim V V V . = X(−1) X ( i1 \ i2 \ ··· \ ii) i=1 {i1,i2,···,ii}⊂{1,2,···,k}

By the induction principle, we know this formula is true for any integer k. ♮ From Theorem 2.4, we get the following additive formula for any two multi-vector spaces.

Corollary 2.3(additive formula) For any two multi-vector spaces V1, V2, e e dim V V dimV dimV dim V V . ( 1 [ 2)= 1 + 2 − ( 1 \ 2) e e e e e e

3. Open problems for a multi- space

Notice that Theorem 2.3 has told us there is a similar linear theory for multi- vector spaces, but the situation is more complex. Here, we present some open problems for further research. Problem 3.1 Similar to linear spaces, define linear transformations on multi-vector spaces. Can we establish a new theory for linear transformations? Problem 3.2 Whether a multi-vector space must be a linear space? Conjecture A There are non-linear multi-vector spaces in multi-vector spaces. Based on Conjecture A, there is a fundamental problem for multi-vector spaces. Problem 3.3 Can we apply multi-vector spaces to non-linear spaces?

References

[1] G.Birkhoff and S.Mac Lane, A Survey of Modern , Macmillan Publishing Co., Inc, 1977. [2] Daniel Deleanu, A Dictionary of Smarandache Mathematics, Buxton University Press, London & New York,2004. [3] K.Hoffman and R.Kunze, (Second Edition), Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1971. [4] L.F.Mao, On Algebraic Multi- Spaces, eprint arXiv: math/0510427, 10/2005. [5] L.F.Mao, Groups of Maps, Surfaces and Smarandache Geome- tries, American Research Press, 2005. [6] F.Smarandache, Mixed noneuclidean , eprint arXiv: math/0010119, 10/2000.

6 [7] F.Smarandache, A Unifying in Logics. Neutrosopy: Neturosophic Proba- bility, Set, and Logic, American research Press, Rehoboth, 1999. [8] F.Smarandache, Neutrosophy, a new Branch of Philosophy, Multi-Valued Logic, Vol.8, No.3(2002)(special issue on Neutrosophy and Neutrosophic Logic), 297- 384. [9] F.Smarandache, A Unifying Field in Logic: Neutrosophic Field, Multi-Valued Logic, Vol.8, No.3(2002)(special issue on Neutrosophy and Neutrosophic Logic), 385-438. [10]] W.B.Vasantha Kandasamy, Bialgebraic structures and Smarandache bialgebraic structures, American Research Press, 2003. [11] W.B.Vasantha Kandasamy and F.Smarandache, Basic Neutrosophic Algebraic Structures and Their Applications to Fuzzy and Neutrosophic Models, HEXIS, Church Rock, 2004.

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