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PREFACE
PREFACE
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FUZZY HV SUBSTRUCTURES IN A TWO DIMENSIONAL
EUCLIDEAN VECTOR SPACE
FUZZY HV SUBSTRUCTURES IN A TWO DIMENSIONAL
EUCLIDEAN VECTOR SPACE
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2
In this paper, we study fuzzy substructures in connection withHvstructures. The original idea comes from geometry, especially from thetwo dimensional Euclidean vector space. Using parameters, we obtain a largenumber of hyperstructures of the grouplike or ringlike types. We connect,also, the mentioned hyperstructures with the thetaoperations to obtain morestrict hyperstructures, as Hvgroups or Hvrings (the dual ones).
1
In this paper, we study fuzzy substructures in connection withHvstructures. The original idea comes from geometry, especially from thetwo dimensional Euclidean vector space. Using parameters, we obtain a largenumber of hyperstructures of the grouplike or ringlike types. We connect,also, the mentioned hyperstructures with the thetaoperations to obtain morestrict hyperstructures, as Hvgroups or Hvrings (the dual ones).
1
9
Achilles
Dramalidis
Achilles
Dramalidis
School of Sciences of Education, Democritus University of
Thrace, 681 00 Alexandroupolis, Greece
School of Sciences of Education, Democritus
Greece
adramali@psed.duth.gr
Thomas
Vougiouklis
Thomas
Vougiouklis
School of Sciences of Education, Democritus University of
Thrace, 681 00 Alexandroupolis, Greece
School of Sciences of Education, Democritus
Greece
tvougiou@eled.duth.gr
Hvstructures
Hvgroup
Fuzzy sets
Fuzzy Hvgroup
[[1] N. Antampoufis, Hypergroups and Hbgroups in complex numbers, Proceedings of 9th AHA##Congress, Journal of Basic Science, Babolsar, Iran, 4(1) (2008), 1725.##[2] P. Corsini and V. Leoreanu, Applications of hyperstructures theory, Kluwer Academic Publishers,##Boston/Dordrecht/London.##[3] B. Davvaz, Fuzzy Hvgroups, Fuzzy Sets and Systems, 101 (1999), 191195.##[4] B. Davvaz, Tfuzzy Hvsubrings of an Hvring, J. Fuzzy Math., 11(1) (2003), 215224.##[5] A. Dramalidis, Dual Hvrings, Rivista di Mathematica Pura ed Applicata, 17 (1996), 5562.##[6] A. Dramalidis, On some classes of Hvstructures, Italian Journal of Pure and Applied Mathematics,##17 (2005), 109114.##[7] A. Dramalidis and T. Vougiouklis, Two fuzzy geometriclike hyperoperations defined on the##same set, 9th AHA, Iran, 2005.##[8] S. Hoskova, Binary hyperstructures determined by relational and transformation systems,##Habilitation thesis, Faculty of Science, University of Ostrava, (2008) 90.##[9] S. Hoskova and J. Chvalina, Abelizations of proximal Hvrings using graphs of good homomorphisms##and diagonals of direct squares of hyperstructures, 8th AHA, Greece, (2003),##[10] S. Hoskova and J. Chvalina, Discrete transformation hypergroups and transformation hypergroups##with phase tolerance space, Discrete Mathematics, 308(18) (2008), 41334143.##[11] L. Konguetsof, Sur les hypermonoides, Bulletin de la Societe Mathematique de Belgique, t.##XXV, 1973.##[12] A. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl., 35 (1971), 512517.##[13] T. Vougiouklis, The fundamental relation in hyperrings: the general hyperfield, Proc. 4th##AHA, World Scientific, (1991), 203211.##[14] T. Vougiouklis, Hyperstructures and their representations, Monographs, Hadronic Press,##USA, 1994.##[15] T. Vougiouklis, A new class of hyperstructures, J. Comb. Inf. Syst. Sciences, 20 (1995),##[16] T. Vougiouklis, The @ hyperoperation, Proceedings: Structure Elements of Hyperstructures,##Alexandroupolis, Greece, (2005), 5364.##[17] T. Vougiouklis, Hvfields and Hvvector spaces with @operations, Proceedings of the 6th##Panhellenic Conference in Algebra and Number Theory, Thessaloniki, Greece, (2006), 95##]
FUZZY PSEUDOTOPOLOGICAL HYPERGROUPOIDS
FUZZY PSEUDOTOPOLOGICAL HYPERGROUPOIDS
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2
On a hypergroupoid one can define a topology such that the hyperoperationis pseudocontinuous or continuous. In this paper we extend thisconcepts to the fuzzy case. We give a connection between the classical and thefuzzy (pseudo)continuous hyperoperations.
1
On a hypergroupoid one can define a topology such that the hyperoperationis pseudocontinuous or continuous. In this paper we extend thisconcepts to the fuzzy case. We give a connection between the classical and thefuzzy (pseudo)continuous hyperoperations.
11
19
Irina
Cristea
Irina
Cristea
DIEA, University of Udine, Via delle Scienze 206, 33100 DIEA, University of Udine, Via delle Scienze 206, 33100 Udine, Italy, Italy
DIEA, University of Udine, Via delle Scienze
Italy
irinacri@yahoo.co.uk
Sarka
Hoskova
Sarka
Hoskova
Department of Mathematics and Physics, University of Defence
Brno, Kounicova 65, 61200 Brno, Czech Republic
Department of Mathematics and Physics, University
Czech Republic
sarka.hoskova@seznam.cz
Hypergroupoid
(Fuzzy) pseudocontinuous hyperoperation
(Fuzzy) continuous hyperoperation
Fuzzy topological space
[[1] R. Ameri, Topological (transposition) hypergroups, Ital. J. Pure Appl. Math., 13 (2003),##[2] C. L. Chang, Fuzzy topological spaces, J. Math. Anal. Appl., 24 (1968), 182190.##[3] P. Corsini, Prolegomena of hypergroup theory, Aviani Editore, 1993.##[4] P. Corsini, Join spaces, power sets, fuzzy sets, Proc. Fifth International Congress on A.H.A.,##1993, Ia¸si, Romania, Hadronic Press, (1994), 4552.##[5] P. Corsini, A new connection between hypergroups and fuzzy sets, Southeast Asian Bull.##Math., 27 (2003), 221229.##[6] P. Corsini, Hyperstructures associated with fuzzy sets endowed with two membership functions,##J. Comb. Inform. Syst. Sci., 31 (2006), 247254.##[7] P. Corsini and V. Leoreanu, Join spaces associated with fuzzy sets, J. Combin. Inform. Syst.##Sci., 20(14) (1995), 293303##[8] P. Corsini and I. Tofan, On fuzzy hypergroups, Pure Math. Appl., 8 (1997), 2937.##[9] P. Corsini and V. Leoreanu, Applications of hyperstructure theory, Kluwer Academic Publishers,##Advances in Mathematics, 2003.##[10] I. Cristea, A property of the connection between fuzzy sets and hypergroupoids, Ital. J. Pure##Appl. Math., 21 (2007), 7382.##[11] I. Cristea, Hyperstructures and fuzzy sets endowed with two membership functions, Fuzzy##Sets and Systems, 160 (2009), 11141124.##[12] I. Cristea, About the fuzzy grade of the direct product of two hypergroupoids, Iran. J. Fuzzy##Syst., to appear.##[13] B. Davvaz, Fuzzy Hvgroups, Fuzzy Sets and Systems, 101 (1999), 191195.##[14] B. Davvaz, Fuzzy Hvsubmodules, Fuzzy Sets and Systems, 117 (2001), 477484.##[15] B. Davvaz and W. A. Dudek, Intuitionistic Hvideals, Int. J. Math. Math. Sci., Art. ID##65921, (2006), 11.##[16] B. Davvaz, W. A. Dudek and Y. B. Jun, Intuitionistic fuzzy Hvsubmodules, Inform. Sci.,##176 (2006), 285300.##[17] B. Davvaz, J. Zhan and K. P. Shum, Generalized fuzzy Hvsubmodules endowed with interval##valued membership functions, Inform. Sci., 178 (2008), 3147315.##[18] W. A. Dudek, J. Zhan and B. Davvaz, On intuitionistic (S, T)fuzzy hypergroups, Soft Computing,##12 (2008), 12291238.##[19] R. Engelking, General topology, PWNPolish Scientific Publishers, Warszawa, (1977), 626.##[20] D. H. Foster, Fuzzy topological groups, J. Math. Anal. Appl., 67 (1979), 549564.##[21] M. Ganster, D. N. Georgiou and S. Jafari, On fuzzy topological groups and fuzzy continuous##functions, Hacet. J. Math. Stat., 34S (2005), 4551##[22] S. Hoskova, Topological hypergroupoids, submitted. ##[23] J. Jantosciak, Transposition hypergroups: noncommutative join spaces, J. Algebra, 187##(1997), 9719.##[24] A. Kehagias and K. Serafimidis, The Lfuzzy Nakano hypergroup, Inform. Sci., 169 (2005),##[25] Y. M. Liu and M. K. Luo, Fuzzy topology, Advances in Fuzzy SystemsApplications and##Theory, World Scientific, 9 (1997).##[26] R. Lowen, Fuzzy topological spaces and fuzzy compactness, J. Math. Anal. Appl., 56 (1976),##[27] J. L. Ma and C. H. Yu, Fuzzy topological groups, Fuzzy Sets and Systems, 12 (1984), 289299.##[28] F. Marty, Sur une generalization de la notion de group, 8th Congress Math. Scandenaves,##Stockholm, (1934), 4549.##[29] A. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl., 35 (1971), 512517.##[30] M. S¸tef˘anescu and I. Cristea, On the fuzzy grade of hypergroups, Fuzzy Sets and Systems,##159(9) (2008), 10971106.##[31] L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338353.##[32] J. Zhan, B. Davvaz and K. P. Shum, On fuzzy isomorphism theorems of hypermodules, Soft##Computing, 11 (2007), 10531057.##[33] J. Zhan, B. Davvaz and K. P. Shum, A new view of fuzzy hypermodules, Acta Math. Sin.##(Engl. Ser.), 23(8) (2007), 13451356.##[34] J. Zhan, B. Davvaz and K. P. Shum, Isomorphism theorems of hypermodules, Acta Math.##Sinica (Chin. Ser.), 50(4) (2007), 909914.##[35] J. Zhan, B. Davvaz and K. P. Shum, A new view of fuzzy hypernearrings, Inform. Sci.,##178(2) (2008), 425438.##[36] J. Zhan and W. A. Dudek, Interval valued intuitionistic (S, T)fuzzy Hvsubmodules, Acta##Math. Sin. (Engl. Ser.), 22 (2006), 963970.##[37] J. Zhan, B. Davvaz and K. P. Shum, A new view of fuzzy hyperquasigroups, J. Intell. Fuzzy##Systems, 20 (2009), 147157.##]
FUZZY HYPERIDEALS IN TERNARY SEMIHYPERRINGS
FUZZY HYPERIDEALS IN TERNARY SEMIHYPERRINGS
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In a ternary semihyperring, addition is a hyperoperation and multiplicationis a ternary operation. Indeed, the notion of ternary semihyperringsis a generalization of semirings. Our main purpose of this paper is to introducethe notions of fuzzy hyperideal and fuzzy bihyperideal in ternary semihyperrings.We give some characterizations of fuzzy hyperideals and investigateseveral kinds of them.
1
In a ternary semihyperring, addition is a hyperoperation and multiplicationis a ternary operation. Indeed, the notion of ternary semihyperringsis a generalization of semirings. Our main purpose of this paper is to introducethe notions of fuzzy hyperideal and fuzzy bihyperideal in ternary semihyperrings.We give some characterizations of fuzzy hyperideals and investigateseveral kinds of them.
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36
Bijan
Davvaz
Bijan
Davvaz
Department of Mathematics, Yazd University, Yazd, Iran
Department of Mathematics, Yazd University,
Iran
davvaz@yazduni.ac.ir, bdavvaz@yahoo.com
Semiring
Semihyperring
Ternary semihyperring
Hyperideal
Subsemihyperring
Fuzzy set
Fuzzy hyperideal
Fuzzy bihyperideal
[[1] R. Ameri and H. Hedayati, On khyperideals of semihyperrings, J. Discrete Math. Sci. Cryptogr.,##10 (2007), 4154.##[2] S. M. Anvariyeh and B. Davvaz, Strongly transitive geometric spaces associated to hypermodules,##Journal of Algebra, 322 (2009), 13401359.##[3] S. M. Anvariyeh, S. Mirvakili and B. Davvaz,  Relation on hypermodules and fundamental##modules over commutative fundamental rings, Communications in Algebra, 36(2) (2008),##[4] S. K. Bhakat and P. Das, Fuzzy subrings and ideals redefined, Fuzzy Sets and Systems, 81##(1996), 383393.##[5] A. Chronowski, On ternary semigroups of lattice homomorphisms, Quasigroups Related Systems,##3 (1996), 5572.##[6] A. Chronowski, Congruences on ternary semigroups, Ukrainian Math. J., 56 (2004), 662681.##[7] P. Corsini, Prolegomena of hypergroup theory, Second edition, Aviani editor, 1993.##[8] P. Corsini and V. Leoreanu, Applications of hyperstructure theory, Advances in Mathematics,##Kluwer Academic Publishers, Dordrecht, 2003.##[9] G. Crombez, On (n,m)rings, Abh. Math. Semin. Univ. Hamburg, 37 (1972), 180199.##[10] G. Crombez and J. Timm, On (n,m)quotient rings, Abh. Math. Semin. Univ. Hamburg, 37##(1972), 200203.##[11] B. Davvaz, Fuzzy Hvgroups, Fuzzy Sets and Systems, 101 (1999), 191195.##[12] B. Davvaz, Fuzzy Hvsubmodules, Fuzzy Sets and Systems, 117 (2001), 477484.##[13] B. Davvaz, Hyperideals in ternary semihyperrings, submitted.##[14] B. Davvaz, P. Corsini and V. LeoreanuFotea, Fuzzy nary subpolygroups, Computers &##Mathematics with Applications, 57 (2009), 141152.##[15] B. Davvaz, P. Corsini and V. LeoreanuFotea, Atanassov’s intuitionistic (S, T)fuzzy nary##subhypergroups and their properties, Information Sciences, 179 (2009), 654666.##[16] B. Davvaz and V. Leoreanu, Applications of interval valued fuzzy nary polygroups with##respect to tnorms (tconorms), Computers & Mathematics with Applications, 57 (2009),##14131424.##[17] B. Davvaz, J. Zhan and K. P. Shum, Generalized fuzzy Hvideals of Hvrings, Int. J. General##Systems, 37(3) (2008), 329346.##[18] B. Davvaz, J. Zhan and K. P. Shum, Generalized fuzzy Hvsubmodules endowed with interval##valued membership functions, Information Sciences, 178(15) (2008), 31473159.##[19] B. Davvaz, Approximations in nary algebraic systems, Soft Computing, 12 (2008), 409418.##[20] B. Davvaz, Rings derived from semihyperrings, Algebras Groups Geom., 20 (2003), 245252.##[21] B. Davvaz, Some results on congruences in semihypergroups, Bull. Malays. Math. Soc., 23(2)##(2000), 5358.##[22] B. Davvaz and V. LeoreanuFotea, Hyperring theory and applications, International Academic##Press, USA, 2007.##[23] B. Davvaz and V. LeoreanuFotea, Binary relations on ternary semihypergroups, Communications##in Algebra, to appear. ##[24] B. Davvaz and A. Salasi, A realization of hyperrings, Communications in Algebra, 34(12)##(2006), 43894400.##[25] B. Davvaz and T. Vougiouklis, Commutative rings obtained from hyperrings (Hvrings) with##relations, Communications in Algebra, 35(11) (2007), 33073320.##[26] B. Davvaz and T. Vougiouklis, nAry hypergroups, Iranian Journal of Science and Technology,##Transaction A, 30(A2) (2006), 165174.##[27] B. Davvaz, W. A. Dudek and S. Mirvakili, Neutral elements, fundamental relations and nary##hypersemigroups, International Journal of Algebra and Computation, 19(4) (2009), 567583.##[28] B. Davvaz, W. A. Dudek and T. Vougiouklis, A generalization of nary algebraic systems,##Communications in Algebra, 37 (2009), 12481263.##[29] V. N. Dixit and S. Dewan, A note on quasi and biideals in ternary semigroups, Internat. J.##Math. & Math. Sci., 18 (1995), 501508.##[30] W. D¨ornte, Untersuchungen ¨uber einen verallgemeinerten Gruppenbegriff, Math. Z., 29##(1928), 119.##[31] W. A. Dudek, Idempotents in nary semigroups, Southeast Asian Bull. Math., 25 (2001),##[32] W. A. Dudek On the divisibility theory in (m, n)rings, Demonstratio Math., 14 (1981),##[33] T. K. Dutta and S. Kar, On regular ternary semirings, Advances in Algebra, Proceedings of##the ICM Satellite Conference in Algebra and Related Topics, World Scientific, New Jersey,##(2003), 343355.##[34] K. Is´eki, Ideals in semirings, Proc. Japan Acad., 34 (1958), 2931.##[35] E. Kasner, An extension of the group concept, Bull. Amer. Math. Soc., 10 (1904), 290291.##[36] O. Kazancı, S. Yamak and B. Davvaz, The lower and upper approximations in a quotient##hypermodule with respect to fuzzy sets, Information Sciences, 178(10) (2008), 23492359.##[37] D. H. Lehmer, A ternary analogue of abelian groups, American J. Math., 59 (1932), 329338.##[38] D. H. Lehmer, A ternary analogue of abelian groups, Amer. J. Math., 59 (1932), 329338.##[39] V. LeoreanuFotea and B. Davvaz, Roughness in nary hypergroups, Information Sciences,##178 (2008), 41144124.##[40] V. LeoreanuFotea and B. Davvaz, nhypergroups and binary relations, European Journal of##Combinatorics, 29 (2008), 10271218.##[41] W. G. Lister, Ternary rings, Trans. Amer. Math. Soc., 154 (1971), 3755.##[42] W. J. Liu, Fuzzy Invariant subgroups and fuzzy ideals, Fuzzy Sets and Systems, 8 (1982),##[43] F. Marty, Sur une generalization de la notion de group, 8th Congress Math. Scandenaves,##Stockholm, (1934), 4549.##[44] V. Maslov, New superposition principle for optimization problems, Sem. sur les Equations##avec D´eriv´ees Partielles 1985/6, Centre Math´ematique de l’Ecole Polytechnique Palaiseau,##esp. 24, 1986.##[45] S. Mirvakili and B. Davvaz, Relations on Krasner (m, n)hyperrings, European Journal of##Combinatorics, to appear.##[46] S. Mirvakili, S. M. Anvariyeh and B. Davvaz, On relation and transitivity conditions of ,##Communications in Algebra, 36 (2008), 16951703.##[47] D. Monk and F. M. Sioson, msemigroups, semigroups, function representations, Fund.##Math., 59 (1966), 233241.##[48] A. Nakassis, Expository and survey article recent results in hyperring and hyperfield theory,##Internat. J. Math. Math. Sci., 11 (1988), 209220.##[49] P. M. Pu and Y. M. Liu, Fuzzy topology I, neighborhood structure of a fuzzy point and##MoorsSmith convergence, J. Math. Anal., 76 (1980), 571599.##[50] E. L. Post, Polyadic groups, Trans. Amer. Math. Soc., 48 (1940), 208350.##[51] F. Poyatos, The JordanH¨older theorem for Asemimodules (Spanish), Rev. Mat. Hisp.##Amer., 32(4) (1972), 251260; ibid. 33(4) (1973) 3648; ibid. 33(4) (1973), 122132.##[52] A. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl., 35 (1971), 512517.##[53] S. A. Rusakov, Some applications of nary group theory, Belaruskaya Navuka, Minsk, 1998. ##[54] M. K. Sen and M. R. Adhikari, On kideals of semirings, Internat. J. Math. & Math. Sci.,##15 (1992), 347350.##[55] F. M. Sioson, Ideal theory in ternary semigroups, Math. Jpn., 10 (1965), 6384.##[56] T. Vougiouklis, Hyperstructures and their representations, Hadronic Press, USA, 1994.##[57] T. Vougiouklis, On some representations of hypergroups, Ann. Sci. Univ. ClermontFerrand##II Math., 26 (1990), 2129.##[58] T. Vougiouklis, The fundamental relation in hyperrings, the general hyperfield, Proc. Fourth##Int. Congress on Algebraic Hyperstructures and Applications (AHA 1990), World Scientific,##(1991), 203211.##[59] X. H. Yuan, C. Zhang and Y. H. Ren, Generalized fuzzy groups and many valued applications,##Fuzzy Sets and Systems, 138 (2003), 205211.##[60] M. M. Zahedi, M. Bolurian and A. Hasankhani, On polygroups and fuzzy subpolygroups, J.##Fuzzy Math., 3 (1995), 115.##]
TFUZZY CONGRUENCES AND TFUZZY FILTERS OF A
BLALGEBRA
TFUZZY CONGRUENCES AND TFUZZY FILTERS OF A
BLALGEBRA
2
2
In this note, we introduce the concept of a fuzzy filter of a BLalgebra,with respect to a tnorm briefly, Tfuzzy filters, and give some relatedresults. In particular, we prove Representation Theorem in BLalgebras. Thenwe generalize the notion of a fuzzy congruence (in a BLalgebra) was definedby Lianzhen et al. to a new fuzzy congruence, specially with respect to a tnorm.We prove that there is a correspondence bijection between the set of allTfuzzy filters of a BLalgebra and the set of all Tfuzzy congruences in thatBLalgebra. Next, we show how Tfuzzy filters induce Tfuzzy congruences,and construct a new BLalgebras, called quotient BLalgebras, and give somehomomorphism theorems.
1
In this note, we introduce the concept of a fuzzy filter of a BLalgebra,with respect to a tnorm briefly, Tfuzzy filters, and give some relatedresults. In particular, we prove Representation Theorem in BLalgebras. Thenwe generalize the notion of a fuzzy congruence (in a BLalgebra) was definedby Lianzhen et al. to a new fuzzy congruence, specially with respect to a tnorm.We prove that there is a correspondence bijection between the set of allTfuzzy filters of a BLalgebra and the set of all Tfuzzy congruences in thatBLalgebra. Next, we show how Tfuzzy filters induce Tfuzzy congruences,and construct a new BLalgebras, called quotient BLalgebras, and give somehomomorphism theorems.
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47
Rajab Ali
Borzooei
Rajab Ali
Borzooei
Department of Mathematics, Shahid Beheshti University, Tehran,
Iran
Department of Mathematics, Shahid Beheshti
Iran
borzooei@sbu.ac.ir
Mahmood
Bakhshi
Mahmood
Bakhshi
Department of Mathematics, Bojnord University, Bojnord, Iran
Department of Mathematics, Bojnord University,
Iran
bakhshi@ub.ac.ir, bakhshimahmood@yahoo.com
Tfuzzy filter
Tfuzzy congruence
[[1] M. Bakhshi and R. A. Borzooei, Lattice structure of fuzzy congruence relations on a hypergroupoid,##Inform. Sci., 177(16) (2007), 33053316.##[2] M. Bakhshi, M. Mashinchi and R. A. Borzooei, Representation of fuzzy structures, International##Review of Fuzzy Mathematics, 1(1) (2006), 7387.##[3] C. C. Chang, Algebraic analysis of many valued logics, Trans. Amer. Math. Soc., 88 (1958),##[4] P. H´ajek, Metamathematics of fuzzy logic, Klower Academic Publishers, Dordrecht, 1998.##[5] E. P. Klement and R. Mesiar, Logical, algebraic, analytic and probabilistic aspects of triangular##norms, Elsevier, Netherlands, 2005.##[6] L. Liu and K. Li, Fuzzy filters of BLalgebras, Inform. Sci., 173 (2005), 141154.##[7] L. Liu and K. Li, Fuzzy Boolean and positive implicative filters of BLalgebras, Fuzzy Sets##and Systems, 152 (2005), 333348.##[8] V. Murali, Fuzzy congruence relations, Fuzzy Sets and Systems., 29 (1984), 391394.##[9] E. Turunen, BLalgebras and basic fuzzy logic, Mathware and Soft Computing, 6 (1999),##[10] E. Turunen, Boolean deductive systems of BLalgebras, Arch. Math. Logic, 40 (2001), 467##]
ON SOME STRUCTURES OF FUZZY NUMBERS
ON SOME STRUCTURES OF FUZZY NUMBERS
2
2
The operations in the set of fuzzy numbers are usually obtained bythe Zadeh extension principle. But these definitions can have some disadvantagesfor the applications both by an algebraic point of view and by practicalaspects. In fact the Zadeh multiplication is not distributive with respect tothe addition, the shape of fuzzy numbers is not preserved by multiplication,the indeterminateness of the sum is too increasing. Then, for the applicationsin the Natural and Social Sciences it is important to individuate some suitablevariants of the classical addition and multiplication of fuzzy numbers that havenot the previous disadvantage. Here, some possible alternatives to the Zadehoperations are studied.
1
The operations in the set of fuzzy numbers are usually obtained bythe Zadeh extension principle. But these definitions can have some disadvantagesfor the applications both by an algebraic point of view and by practicalaspects. In fact the Zadeh multiplication is not distributive with respect tothe addition, the shape of fuzzy numbers is not preserved by multiplication,the indeterminateness of the sum is too increasing. Then, for the applicationsin the Natural and Social Sciences it is important to individuate some suitablevariants of the classical addition and multiplication of fuzzy numbers that havenot the previous disadvantage. Here, some possible alternatives to the Zadehoperations are studied.
49
59
Antonio
Maturo
Antonio
Maturo
Department of Social Sciences, University of ChietiPescara, via
dei Vestini, 66013, Chieti, Italia
Department of Social Sciences, University
Italy
amasturo@unich.it
Fuzzy numbers
Fuzzy algebraic structures
Alternative fuzzy operations
Fuzzy hyperoperations
[[1] B. Bede and J. Fodor, Product type operations between fuzzy numbers and their applications##in Geology, Acta Polytechnica Hungarica, 3(1) (2006), 123139.##[2] P. Corsini, Prolegomena of hypergroup theory, Aviani Editore, Tricesimo, 1993.##[3] P. Corsini and V. Leoreanu, Applications of hyperstructure theory, Kluwer Academic Publishers,##Dordrecht, Hardbound, 2003.##[4] B. De Finetti, Theory of probability, J. Wiley, New York, 12 (1974).##[5] D. Dubois and H. Prade, Fuzzy numbers: an overview, in J. C. Bedzek and Ed. Analysis of##Fuzzy Information , CRCPress, Boca Raton, 2 (1988), 339.##[6] S. Hoskova, Binary hyperstructures determined by relational and transformation systems,##Habilitation Thesis, Faculty of Science, University of Ostrava, (2008), 90.##[7] S. Hoskova and J. Chvalina, Discrete transformation hypergroups and transformation hypergroups##with phase tolerance space, Discrete Mathematics, 308(18) (2008), 41334143.##[8] G. Klir and B. Yuan, Fuzzy sets and fuzzy logic: theory and applications, Prentice Hall, New##Jersey, 1995.##[9] M. Mares, Weak arithmetic on fuzzy numbers, Fuzzy Sets and Systems, 91(2) (1997), 143##[10] A. Maturo, Grandezze aleatorie fuzzy e loro previsioni per le decisioni in condizione di informazione##parziale, Current Topics in Computer Sciences, Cortellini and Luchian Ed., Panfilus,##Iasi, (2004), 1524.##[11] A. Maturo, Fuzzy conditional probabilities by the subjective point of view, Advances in Mathematics##of Uncertainty, Tofan Ed., Performantica, Iasi, (2006), 99108.##[12] A. Maturo, Fuzzy events and their probability assessments, Journal of Discrete Mathematical##Sciences and Cryptography, 3(13) (2000), 8394.##[13] A. Maturo, Alternative fuzzy operations and applications to social sciences, International##Journal of Intelligent Systems, to appear, printed on line by Wiley, 2009.##[14] A. Maturo and A. Ventre, On some extensions of the de Finetti coherent prevision in a fuzzy##ambit, Journal of Basic Science, 4(1) (2008), 95103.##[15] M. Squillante and A. G. S. Ventre, Consistency for uncertainty measure, International Journal##of Intelligent Systems, 13 (1998), 419430.##[16] M. Sugeno, Theory of fuzzy integral and its applications, Ph. D. Thesis, Tokyo, 1974.##[17] S. Weber, Decomposable measures and integrals for archimedean tconorms, J. Math. Anal.##Appl., 101(1) (1984), 114138.##[18] R. Yager, A characterization of the extension principle, Fuzzy Sets and Systems, 18(3)##(1986), 205217.##[19] L. Zadeh, Fuzzy sets, Inf. Control, 8 (1965), 338353.##[20] L. Zadeh, Probability measures of fuzzy events, J. Math. Anal. Appl., 23 (1968), 421427.##[21] L. Zadeh, The concept of a linguistic variable and its application to approximate reasoning##I, II, III, Information Sciences, 8 (1975), 199249 and 301357, 9 (1975), 4380.##]
SPECTRUM OF PRIME FUZZY HYPERIDEALS
SPECTRUM OF PRIME FUZZY HYPERIDEALS
2
2
Let R be a commutative hyperring with identity. We introduceand study prime fuzzy hyperideals of R. We investigate the Zariski topologyon FHspec(R), the spectrum of prime fuzzy hyperideals of R.
1
Let R be a commutative hyperring with identity. We introduceand study prime fuzzy hyperideals of R. We investigate the Zariski topologyon FHspec(R), the spectrum of prime fuzzy hyperideals of R.
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Commutative hyperring
Prime fuzzy hyperideals
Zariski topology
[[1] R. Ameri and R. Mahjoob, Prime spectrum of Lsubmodules, Fuzzy Sets and Systems, 159(9)##(2008), 11071115.##[2] R. Ameri and R. Mahjoob, Zariski topology on the spectrum of prime Lsubmodules, Soft##Computing, 12(9) (2008), 901908.##[3] R. Ameri and N. Shafiiyan, Fuzzy prime and primary ideals of hyperrings, to appear.##[4] S. K. Bhambri, R. Kumar and P. Kumar, Fuzzy prime submodules and radical of a fuzzy##submodules, Bull. Cal. Math. Soc., 87 (1993), 163168.##[5] P. Corsini, Prolegomena of hypergroup theory , Second Edition Aviani Editer, 1993.##[6] V. N. Dixit, R. Kummar and N. Ajmal, Fuzzy ideals and fuzzy prime ideals of a ring, Fuzzy##Sets and Systems, 44 (1991), 127138.##[7] J. A. Goguen, Lfuzzy sets, Journal Math. Appl., 18 (1967), 145174.##[8] H. HadjiAbadi and M. M. Zahedi, Some results on fuzzy prime spectrum of a ring, Fuzzy##Sets and Systems, 77 (1996), 235240.##[9] R. Kumar, Fuzzy prime spectrum of a ring, Fuzzy Sets and Systems, 46 (1992), 147154.##[10] R. Kumar and J. K. Kohli, Fuzzy prime spectrum of a ring II, Fuzzy Sets and Systems, 59##(1993), 223230.##[11] H. V. Kumbhojkar, Some comments on spectrum of prime fuzzy ideals of a ring, Fuzzy Sets##and Systems, 85 (1997), 109114.##[12] H. V. Kumbhojkar, Spectrum of prime fuzzy ideals, Fuzzy Sets and Systems, 62 (1994),##[13] C . P. Lu, Prime submodules of modules, Comm. Math. Univ., 33 (1987), 6169.##[14] C. P. Lu, The Zariski topology on the spectrum of a modules, Houston Journal of Mathematics,##25(3) (1999), 417432.##[15] C. P. Lu, Spectra of modules, Comm. in Algebra, 23(10) (1995), 37413752.##[16] F. Marty, Su rune generalization de la notion de groupe, 8th Congress Math. Scandinaves,##Stokholm, (1934), 4549.##[17] C. G. Massouros, Free and cyclic hypermodules, Annali di Matematica Pura ed Applicata, 4##(1988), 153166.##[18] R. L. McCasland, M. E. Moore and P. F. Smith, On the spectrum of modules over a commutative##ring, Communications in Algebra, 25(1) (1997), 79103.##[19] J. N. Mordeson and D. S. Malik, Fuzzy commutative algebra, World Scientific Publishing Co.##Pet. Ltd, 1998.##[20] T. K. Mukherjee and M. K. Sen, On fuzzy ideals of a ring I, Fuzzy Sets and Systems, 21##(1987), 99104.##[21] C. V. Negoita and D. A. Ralescu, Application of fuzzy systems analysis, Birkhauser, Basel,##[22] R. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl., 35 (1971), 512517.##[23] F. I. Sidky, On radical of fuzzy submodules and primary fuzzy submodules, Fuzzy Sets and##Systems, 119 (2001), 419425.##[24] L. A. Zadeh, Fuzzy sets, Inform and Control, 8 (1965), 338353.##[25] F. Z. Pan, Fuzzy finitely generated modules, Fuzzy Sets and Systems, 21 (1987), 105113.##]
SOME PROPERTIES OF TFUZZY GENERALIZED SUBGROUPS
SOME PROPERTIES OF TFUZZY GENERALIZED SUBGROUPS
2
2
In this paper, we deal with Molaei’s generalized groups. We definethe notion of a fuzzy generalized subgroup with respect to a tnorm (orTfuzzy generalized subgroup) and give some related properties. Especially,we state and prove the Representation Theorem for these fuzzy generalizedsubgroups. Next, using the concept of continuity of tnorms we obtain a correspondencebetween TF(G), the set of all Tfuzzy generalized subgroups of ageneralized group G, and the set of all Tfuzzy generalized subgroups of thecorresponding quotient generalized group. Subsequently, we study the quotientstructure of Tfuzzy generalized subgroups: we define the notion of aTfuzzy normal generalized subgroup, give some related properties, constructthe quotient generalized group, state and prove the homomorphism theorem.Finally, we study the lattice of Tfuzzy generalized subgroups and prove thatTF(G) is a Heyting algebra.
1
In this paper, we deal with Molaei’s generalized groups. We definethe notion of a fuzzy generalized subgroup with respect to a tnorm (orTfuzzy generalized subgroup) and give some related properties. Especially,we state and prove the Representation Theorem for these fuzzy generalizedsubgroups. Next, using the concept of continuity of tnorms we obtain a correspondencebetween TF(G), the set of all Tfuzzy generalized subgroups of ageneralized group G, and the set of all Tfuzzy generalized subgroups of thecorresponding quotient generalized group. Subsequently, we study the quotientstructure of Tfuzzy generalized subgroups: we define the notion of aTfuzzy normal generalized subgroup, give some related properties, constructthe quotient generalized group, state and prove the homomorphism theorem.Finally, we study the lattice of Tfuzzy generalized subgroups and prove thatTF(G) is a Heyting algebra.
73
87
Mahmood
Bakhshi
Mahmood
Bakhshi
Department of Mathematics, University of Bojnord, Bojnord,
Iran
Department of Mathematics, University of
Iran
bakhshi@ub.ac.ir, bakhshimahmood@yahoo.com
Rajab Ali
Borzooei
Rajab Ali
Borzooei
Department of Mathematics, Shahid Beheshti University, Tehran,
Iran
Department of Mathematics, Shahid Beheshti
Iran
borzooei@sbu.ac.ir
Generalized groups
Fuzzy generalized subgroups
tnorm
Heyting algebra
[[1] R. Ameri, M. Bakhshi, S. A. Nematollahzadeh and R. A. Borzooei, Fuzzy (strong) congruence##relations on a hypergroupoid and hyper BCKalgebra, Quasigroups and Related Systems, 15##(2007), 1124.##[2] J. M. Anthony and H. Sherwood, A characterization of fuzzy subgroups, Fuzzy Sets and##Systems, 7 (1982), 297305.##[3] M. Bakhshi and R. A. Borzooei, Lattice structure on fuzzy congruence relations of a hypergroupoid,##Inform. Sci., 177(16) (2007), 33053313.##[4] G. Birkhoff, Lattice theory, Amer. Math. Soc., Providence, R. I., 1967.##[5] R. A. Borzooei, M. Bakhshi and M. Mashinchi, Lattice structure on some fuzzy algebraic##systems, Soft Computing, 12(8) (2008), 739749.##[6] R. A. Borzooei, G. R. Rezaei, M. R. Molaei and M. M. Zahedi, Characterization of generalized##groups of orders 2 and 3, Pure Math. Appl., 11(4) (2000), 174.##[7] E. P. Klement and R. Mesiar, Logical, algebraic, analytic and probabilistic aspects of triangular##norms, Elsevier, Netherlands, 2005. ##[8] W. J. Liu, Fuzzy invariant subgroups and fuzzy ideals, Fuzzy Sets and Sysems, 8 (1982),##[9] M. Mehrabi, M. R. Molaei and A. Oloomi, Generalized subgroups and homomorphisms, Arab##J. Math. Sci., 6(2) (2000), 17.##[10] M. R. Molaei, Generalized groups, Buletinul Institului Polithnic Din Iasi, Tomul XLV(XLIX),##(1999), 2124.##[11] J. N. Mordeson, K. R. Bhutani and A. Rosenfeld, Fuzzy group theory, SpringerVerlag Berlin##Heidelberg, Netherlands, 2005.##[12] N. P. Mukherjee and P. Bhattacharya, Fuzzy normal subgroups and fuzzy cosets, Inform. Sci.,##34 (1984), 225239.##[13] M. T. Abu Osman, On some product of fuzzy subgroups, Fuzzy Sets and Systems, 24 (1987),##[14] H. T. Nguyen and E. A.Walker, A first course in fuzzy logic, 3rd ed. Chapman and Hall/CRC,##USA, Florida, 2006.##[15] E. H. Roh, B. Davvaz and K. H. Kim, Tfuzzy subhypernearrings of hypernearrings, Sci.##Math. Jpn., 61(3) (2005), 535545.##[16] A. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl., 35 (1971), 512517.##[17] S. Sessa, On fuzzy subgroups and fuzzy ideals under triangular norms: short communiction,##Fuzzy Sets and Systems, 13 (1984), 95100.##[18] L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338353.##[19] J. Zhang, On properties of fuzzy hyperideals in hypernearrings with tnorms, J. Appl. Math.##Comput., 20 (2006), 255277.##]