- Email: [email protected]

Chaos, Solitons and Fractals 40 (2009) 2146–2151 www.elsevier.com/locate/chaos

On the classiﬁcation of fuzzy projective planes of fuzzy 3-dimensional projective space S. Ekmekc¸i, Z. Akc¸a *, A. Bayar Eskisßehir Osmangazi University, Department of Mathematics, 26480 Eskisßehir, Turkey Accepted 1 October 2007

Abstract In this work, the classiﬁcations of fuzzy 3-dimensional vector spaces of fuzzy 4-dimensional vector space and fuzzy projective planes of fuzzy 3-dimensional projective space from fuzzy 4-dimensional vector space are given. 2007 Elsevier Ltd. All rights reserved.

1. Introduction and preliminaries Projective geometry provides a geometric way to study vector spaces. Indeed, a projective space over a skew ﬁeld K is nothing else than the lattice of proper non-trivial subspaces of a vector space over K. This is the origin of projective geometry [3,5]. There is a physical bridge between physical space and 3-dimensional projective space. It is well known that a lot of work in physics involves just the directions of vectors and the directions of various quantities. Thinking of four-dimensional space is something that is hard to imagine, but we cannot really imagine it. When we are concerned just with directions, the things in the space of physics, we can represent them all in terms of three-dimensional space according to the methods of projective geometry. El Naschie MS. [4] introduced an egg example to illustrate an intriguing practical application of the fact that by limiting spatial extension a suﬃciently small area of a curved surface is used for practical purposes, ﬂat in structural engineering. We have a 3-dimensional projective space in which there is an absolute quadric; we imagine this quadric to be a sort of ellipsoid and any direction in the physical space corresponds to a point in this 3-dimensional projective space. We see, at once, that there are three kinds of points: points that lie inside the ellipsoid, points that lie outside the ellipsoid and points that lie on the ellipsoid. Now, these correspond to the three kinds of directions which we have in physical space. Points inside the quadric correspond to direction of the particle-timeline direction: points outside the quadric correspond to space line directions; points on the quadrant correspond to directions of the light rays, or null directions, as you might call them. These directions are all immediately pictured in the three-dimensional projective space, and all the other relationships between directions of physical space can immediately be in physical space.

*

Corresponding author. Tel.: +90 0 222 2291715; fax: +90 0 222 2393578. E-mail addresses: [email protected] (S. Ekmekc¸i), [email protected] (Z. Akc¸a), [email protected] (A. Bayar).

0960-0779/$ - see front matter 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2007.10.002

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We now deﬁne the n-dimensional projective space PG(n, K) for n > 0 and K, any (skew) ﬁeld. Let V be any vector space of dimension n + 1 over K. Then PG(n, K), the n-dimensional projective space over K, is the set of all subspaces of V distinct from the trivial subspaces fog and V. The 1-dimensional subspaces are called the points of PG(n, K), the 2dimensional subspaces are called the (projective) lines and the 3-dimensional ones are called (projective) planes. We remark that by going from a vector space to the associated projective space, the dimension drops by one unit. Hence an (n + 1)-dimensional vector space gives rise to an n-dimensional projective space [3]. Let us give an example to ﬁx the ideas. Let K be any ﬁeld, and let V be a 3-dimensional vector space over K. The projective plane PG(V), also denoted by PG(2, K) because it has dimension 2, consists of all 1- and 2-dimensional subspaces of V (these are indeed exactly the proper nontrivial subspaces of V). For dimension reasons we call the 1-dimensional subspaces points and the 2-dimensional ones lines. Note the following properties of PG(V). Considering two points, A and B, there is a unique line L containing both A and B. Indeed, A and B are just two diﬀerent 1-dimensional subspaces of V, and there is exactly one 2-dimensional subspace L spanned by A and B. Also, considering two lines, L and M, there is a unique point A contained in both L and M. Indeed, L and M are two diﬀerent 2-dimensional subspaces of V, and these intersect in a unique 1-dimensional subspace A by dimension considerations. Finally, there are four points in such a position that they pairwise deﬁne six diﬀerent lines. Indeed, we can take the vector lines generated by the vectors with coordinates (1, 0, 0), (0, 1, 0), (0, 0, 1) and (1, 1, 1), respectively (after introducing coordinates in V). Projective spaces have been fuzzyﬁed by Kuijken et al., see [6]. Our aim is now to give the classiﬁcation of fuzzy 3-dimensional vector spaces of fuzzy 4-dimensional vector space and then the classiﬁcation of the fuzzy projective planes of fuzzy 3-dimensional projective space from fuzzy 4-dimensional vector space. In this paper V will always denote a four-dimensional vector space over some ﬁeld, K, o being the origin of V. We denote the minimum operator by ^ and the maximum operator by _. We consider fuzzy sets having membership degrees in the unit interval [0, 1]. Classical subspaces of V will be denoted Ui. The following deﬁnitions and theorems concerning the basic concepts of the subject has been taken from [1,2,6–10] with some small modiﬁcations. Deﬁnition 1.1. Let k : V ! [0,1] be a fuzzy set on V. Then we call k a fuzzy vector space on V if and only if kða u þ b vÞ P kðuÞ ^ kðvÞ 8u; v 2 V and "a, b 2 K. Proposition 1.1. Let V be a vector space over some field K, u; v 2 V and a 2 Kn{0}. If k : V ! [0, 1] is a fuzzy vector space, then we have (i) kða uÞ ¼ kðuÞ; (ii) kðoÞ ¼ supu2V kðuÞ; (iii) if kðuÞ–kðvÞ, we have kðu þ vÞ ¼ kðuÞ ^ kðvÞ.

Deﬁnition 1.2. Let k be a fuzzy vector space on V. The subspace L (linearly), generated by Sup p(k), Sup p(k) = {x 2 V : k(x) = 0}, is called the base vector space of k. The dimension d(k) of a fuzzy vector space of V is the dimension of its base subspace. Deﬁnition 1.3. If U is an i-dimensional subspace of V, and (k, U) is a fuzzy vector space, then it is called a fuzzy idimensional vector space on U. If i = 1, i.e. U is a vector line, then (k, U) is a fuzzy vector line on U, if i = 2, i.e. U is a plane, (k, U) will be called a fuzzy vector plane on U. If i = n 1, then (k, U) is called a fuzzy vector hyperplane on U. Let V be an n-dimensional vector space over some ﬁeld K, with n P 2. Let L be a vector line of V, so L is uniquely deﬁned by some nonzero vector u. Let a be a vector plane of the n-dimensional vector space V (n P 3), then we know that a is uniquely deﬁned by two linearly independent vectors, u and v. Theorem 1.1. If k : L ! [0, 1] is a fuzzy vector line on L, then kðuÞ ¼ kðvÞ 8u; v 2 L n fog, and kðoÞ P kðuÞ 8u 2 L. Theorem 1.2. If k : a ! [0, 1] is a fuzzy vector plane on a, then there exists a vector line L of a and real numbers a0 P a1 P a2 2 [0, 1] such that k is of the following form:

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k:

a ! ½0; 1 o ! a0 u ! a1 for u 2 L n fog u ! a2 for u 2 a n L:

Deﬁnition 1.4. Suppose V is an n-dimensional vector space. A ﬂag in V is a sequence of distinct, non-trivial subspaces (U0, U1, . . ., Um) such that Uj Ui for all j < i < n 1. The rank of a ﬂag is the number of subspaces it contains. A maximal ﬂag in V is a ﬂag of length n.

2. Fuzzy 3-dimensional vector spaces of fuzzy 4-dimensional vector spaces In this work, we now classify fuzzy 3-dimensional vector subspaces of fuzzy 4-dimensional vector spaces to classify fuzzy projective planes of fuzzy 3-dimensional projective space. Since a subspace should not necessarily have the same values (membership degrees diﬀerent from a0) in its points as the whole space [6], this classiﬁcation is given in the following theorem. Theorem 2.1. Let V be a 4-dimensional vector space over some field K and k : V ! [0, 1] be a fuzzy vector space on V. Then the fuzzy 4-dimensional vector space k has exactly four kinds of fuzzy 3-dimensional vector spaces. Proof. Let k : V ! [0, 1] be a fuzzy vector space on V and (U0, U1, U2, U3, V) be a maximal ﬂag, then there exists a vector plane a of V and a base line L of a and real numbers a0 P a P b P c P d 2 [0, 1] such that k:

V ! ½0; 1 o ! a0 u ! a for u 2 U 1 n fU 0 g u ! b for u 2 U 2 n U 1 u ! c for u 2 U 3 n U 2 u ! d for u 2 V n U 3 :

The number of points diﬀerent from zero on a vector line is denoted by p, q counts the number of vector lines Lj passing through zero point different from base line in U2, r counts the number of vector lines Lk passing through zero point different from base line in U3nLj and s counts the number of vector lines Lt in VnU3. These fuzzy vector planes of k are one of the following forms: (1) Let ajk be 3-dimensional vector spaces, kijk :

ajk ! ½0; 1 o ! a0 u ! ai for u 2 L n fog u ! bij for u 2 U 2 n L u ! cik

for u 2 U 3 n L

such that ai P bij P cik, i 2 {1, . . ., p}, j 2 {1, . . ., q}, k 2 {1, . . ., r}. (2) Let bjt be 3-dimensional vector spaces, with lijt :

bjt ! ½0; 1 o ! a0 u ! ai for u 2 L n fog u ! bij for u 2 U 2 n L u ! d it for u 2 V n U 3

such that ai P bij P dit, i 2 {1, . . ., p}, j 2 {1, . . ., q}, t 2 {1, . . ., s}.

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(3) Let ckt be 3-dimensional vector spaces, mikt :

ckt ! ½0; 1 o ! a0 u ! ai for u 2 L n fog u ! cik for u 2 U 3 n U 2 u ! d it for u 2 V n U 3

such that ai P cik P dit, i 2 {1, . . ., p}, k 2 {1, . . ., r}, t 2 {1, . . ., s}. (4) Let djkt be 3-dimensional vector spaces, nijkt :

djkt ! ½0; 1 o ! a0 u ! bij u ! cik u ! d it

for u 2 Lj n fog for u 2 U 3 n U 2 for u 2 V n U 3

such that bij P cik P dit, i 2 {1, . . ., p}, j 2 {1, . . ., q}, k 2 {1, . . ., r}, t 2 {1, . . ., s}. Any fuzzy vector plane is in one of the four classes.

h

3. Fuzzy projective planes of fuzzy 3-dimensional projective space A general deﬁnition of fuzzy n-dimensional projective space k 0 is well known [6]. Here, we restrict ourselves to the case a fuzzy 3-dimensional projective space k 0 from a fuzzy 4-dimensional vector space (k, V) having the following form: k:

V ! ½0; 1 o ! a0 u ! a for u 2 U 1 n fU 0 g u ! b for u 2 U 2 n U 1 u ! c for u 2 U 3 n U 2 u ! d for u 2 V n U 3

ð1Þ

with Ui an i-dimensional subspace of V, containing all Uj for j < i, and a0 P a P b P c P d are reals in [0, 1]. We deﬁne a fuzzy 3-dimensional projective space k 0 on V 0 as follows, where it will be denoted FPG(3, K) k0 :

V 0 ! ½0; 1 q!a p ! b for p 2 U 01 n fqg p ! c for p 2 U 02 n U 01 p ! d for p 2 V 0 n U 02

ð2Þ

with q the fuzzy projective point corresponding to the fuzzy vector line U1 in (2) and U 0i the i-dimensional projective space, corresponding to the vector space Ui+1. Then, the sequence ðq; U 01 ; U 02 ; V 0 Þ is a maximal ﬂag and a P b P c P d are reals in [0, 1]. The following theorem deals with the classiﬁcation of fuzzy projective lines of fuzzy 3-dimensional projective space from fuzzy 4-dimensional vector space. Theorem 3.1. Fuzzy 3-dimensional projective space k 0 from fuzzy 4-dimensional vector space k over some field K has exactly four kinds of fuzzy projective planes. Proof. Let k 0 be fuzzy 3-dimensional projective space on V 0 . Then it is as follows:

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k0 :

V 0 ! ½0; 1 q!a p ! b for p 2 U 01 n fqg p ! c for p 2 U 02 n U 01 p ! d for p 2 V 0 n U 02 :

The fuzzy projective planes of k 0 are one of the following forms. (1) Let Kjk be the projective planes corresponding to the vector spaces ajk, and q be the projective point corresponding to the vector line L k0ijk :

K jk ! ½0; 1 q ! ai p ! bij ; for p 2 U 01 n fqg p ! cik for p 2 U 02 n U 01

such that ai P bij P cik. (2) Let Ljt be the projective planes corresponding to the vector spaces bjt, and q be a projective point corresponding to the vector line L l0ijt :

Ljt ! ½0; 1 q ! ai p ! bij ; for p 2 U 01 n fqg p ! d it for p 2 V 0 n U 02

such that ai P bij P dit. (3) Let Mkt be the projective planes corresponding to the vector spaces ct, and q be a projective point corresponding to the vector line L m0ikt :

M kt ! ½0; 1 q ! ai p ! cik ; for p 2 U 02 n U 01 p ! d it for p 2 V 0 n U 02

such that ai P cik P dit. (4) Let Njk be the projective planes corresponding to the vector spaces ajk n0ijkt :

N jkt ! ½0; 1 p ! bij for p 2 U 01 n fqg p ! cik ; for p 2 U 02 n U 01 p ! d it for p 2 V 0 n U 02

such that bij P cik P dit. One can easily see that any fuzzy projective plane is in one of the above four classes.

h

4. Conclusion The complex n-space Cn with the metric d s2 ¼

n X

dza dza

a¼1

(where z1, z2, . . ., zn is the natural coordinate system) is a complete, ﬂat Kaehler manifold. If we take n = 2, then ðR4 ; g; jÞ is a Kaehler manifold, where

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g : gððx1 ; y 1 ; x2 ; y 2 Þ; ðu1 ; v1 ; u2 ; v2 ÞÞ ¼ x1 u1 þ y 1 v1 þ x2 u2 þ y 2 v2 is an Euclidean metric and j : jðx1 ; y 1 ; x2 ; y 2 Þ ¼ ðy 1 ; x1 ; y 2 ; x2 Þ is a canonic complex structure. In the present paper, we are dealing with fuzzy 3-dimensional projective space k 0 from a fuzzy 4-dimensional vector space (k, V). If one choses V ¼ R4 in Theorem 2.1 then the Kaehler manifold ðR4 ; g; jÞ can be fuzzyﬁed. In a forthcoming work, we investigate a possible connection between El Naschie’s research on 4-dimensional Fuzzy Kaehler Manifold and fuzzy 3-dimensional projective space k 0 from a fuzzy 4-dimensional vector space (k, V).

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