International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No1, January 2013 DOI : 10.5121/ijfls.2013.3106 55 Lattices of Fuzzy Measures Defined on Hilbert Spaces Manju Cherian and K.Sudheer Associate Professor,Department of Mathematics, Farook College,Kozhikode, Kerala-673 632 India. [email protected]Abstract A new type of translation invariant and lower semi continuous fuzzy measure, called VGFM (Vector Generated Fuzzy Measure), on the class of subsets of a real Hilbert space is introduced. It measures a subset of the Hilbert space as a projection of the set along a given unit vector in the Hilbert space. Any VGFM uniquely de_nes two orthogonal subspaces called the support space and null space whose direct sum is the Hilbert space. An equivalence of VGFMs is defined related to the support space. A partial order relation similar to that of absolute continuity is defined on the class of all VGFMs. It is proved that this partial order makes the class of all VGFMs a lattice. Further properties of the lattice of VGFMs are studied. Key words : Vector Generated Fuzzy Measure, Support space, Null space,Orthogonal VGFMs, Absolute Continuity, Lattice, Join, Meet, Atomic VGFM,Finite Dimensional VGFM. 1. Introduction A new type of fuzzy measure on a real Hilbert space was de_ned [6] by Manju Cherian and K. Sudheer. It was proved [7] that the fuzzy measure of a compact and convex set resembles the concept of length of an intervalas the di_erence between the end points. An equivalence relation based on the support space of the VGFM is de_ned. On the class of all VGFMs on a Hilbert space, a partial order relation is introduced. This partial order is compared with the absolute continuity of classical measures. It is proved that the partial order makes the class of VGFMs a lattice.
Transcript
International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No1, January 2013
DOI : 10.5121/ijfls.2013.3106 55
Lattices of Fuzzy Measures Defined on HilbertSpaces
Manju Cherian and K.Sudheer
Associate Professor,Department of Mathematics,Farook College,Kozhikode, Kerala-673 632
A new type of translation invariant and lower semi continuous fuzzy measure, called VGFM (VectorGenerated Fuzzy Measure), on the class of subsets of a real Hilbert space is introduced. It measures asubset of the Hilbert space as a projection of the set along a given unit vector in the Hilbert space. AnyVGFM uniquely de_nes two orthogonal subspaces called the support space and null space whose directsum is the Hilbert space. An equivalence of VGFMs is defined related to the support space. A partial orderrelation similar to that of absolute continuity is defined on the class of all VGFMs. It is proved that thispartial order makes the class of all VGFMs a lattice. Further properties of the lattice of VGFMs arestudied.
A new type of fuzzy measure on a real Hilbert space was de_ned [6] by Manju Cherian and K.Sudheer. It was proved [7] that the fuzzy measure of a compact and convex set resembles theconcept of length of an intervalas the di_erence between the end points. An equivalence relationbased on the support space of the VGFM is de_ned. On the class of all VGFMs on a Hilbertspace, a partial order relation is introduced. This partial order is compared with the absolutecontinuity of classical measures. It is proved that the partial order makes the class of VGFMs alattice.
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2. Preliminaries
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3. Lattice of VGFMs
Precedence is introduced among the class of all VGFMs. It is proved that this resembles theabsolute continuity of fuzzy measures. Moreover it forms a partial order making the set of allVGFMs a lattice. Precedence is given the same notation as that of absolute continuity of fuzzymeasures.
Proof. The conditions of reflexivity, symmetry and transitivity follow directly from the definition.
Note3.3. Under the equivalence defined, the collection of all VGFMs becomes an equivalenceclass. Each member of this class represents a collection of VGFMs equivalent to it. If the Hilbertspace is assumed to be separable, it is possible to have a VGFM corresponding to everyorthogonal pair of subspaces making a decomposition of the Hilbert.
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Theorem3.4. For each pair of non-empty subspaces of a Hilbert space which are orthogonalcomplements of each other, there is a VGFM for which these two subspaces chosen arerespectively the support and null spaces.
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References
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[7] Manju Cherian & K. Sudheer, Sets Having Finite Fuzzy Measure in Real Hilbert Spaces, The Journalof Fuzzy Mathematics Vol. 20, No. 1, 2012.
[8] Mila Stojokovic, Zoran Stojokovic, Integral With Respect To Fuzzy Measure In Finite DimensionalBanach Spaces, Novi Sad J. Math. Vol. 37, No. 1, 2007. 163 - 170.
[9] Walter Rudin, Real and Complex Analysis, T M H Publishing Company Ltd. New Delhi, 1966.[10] Zhenyuan Wang and George J. Klir, Fuzzy Measure Theory, Plenum Press, NewYork and London
