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$L^M$-valued equalities, $L^M$-rough approximation operators and $ML$-graded ditopologies

Year 2017, Volume: 46 Issue: 1, 15 - 32, 01.02.2017

Abstract

We introduce a certain many-valued generalization of the concept of an $L$-valued equality called an $L^M$-valued equality. Properties of $L^M$-valued equalities are studied and a construction of an $L^M$-valued equality from a pseudo-metric is presented. $L^M$-valued equalities are applied to introduce upper and lower $L^M$-rough approximation operators, which are essentially many-valued generalizations of Z. Pawlak's rough approximation operators and of their fuzzy counterparts. We study properties of these operators and their mutual interrelations. In its turn, $L^M$-rough approximation operators are used to induce topological-type structures, called here $ML$-graded ditopologies.

References

  • U. Bodenhofer, Ordering of fuzzy sets based on fuzzy orderings. I: The basic approach, Mathware Soft Comput. 15 (2008) 201218.
  • U. Bodenhofer, Ordering of fuzzy sets based on fuzzy orderings. II: Generalizations, Mathware Soft Comput. 15 (2008) 219249.
  • L.M. Brown, M. Diker, Ditopological texture spaces and intuitionistic sets Fuzzy Sets and Syst. 98, (1998) 217224.
  • L.M. Brown, R. Ertürk, “S. Dost, Ditopological texture spaces and fuzzy topology, I. Basic concepts, Fuzzy Sets and Syst. 147, (2004) 171199.
  • L.M. Brown, R. Ertürk, “S. Dost, Ditopological texture spaces and fuzzy topology, II. Topo- logical considerations, Fuzzy Sets and Syst. 147, (2004) 201231. 18861912.
  • L.M. Brown, A. ’Sostak, Categories of fuzzy topologies in the context of graded ditopologies, Iranian J. Of Fuzzy Systems, Systems 11, No. 6, (2014) pp. 1-20
  • C.L. Chang, Fuzzy topological spaces, J. Math. Anal. Appl. 24, 182190 (1968)
  • K.C. Chattopadhyay, R.N. Nasra, S.K. Samanta, Gradation of openness: Fuzzy topology, Fuzzy Sets and Syst. 48 (1992), 237242.
  • K.C. Chattopadhyay, S.K. Samanta, Fuzzy closure operators, fuzzy compactness and fuzzy connectedness, Fuzzy Sets and Syst. 54 (1992), 237242.
  • P. Chen, D. Zhang, Alexandro L-cotopological spaces, Fuzzy Sets and Systems 161 (2010) 2505  2525.
  • M. Demirci, Fuzzy functions and their fundamental properties, Fuzzy Sets Syst. 106 (1999), 239246
  • D. Dubois, H. Prade, Rough fuzzy sets and fuzzy rough sets. Internat. J. General Syst. 17, 191209 (1990)
  • A. Elkins, A. ’Sostak, On some categories of approximate systems generated by L-relations. In: 3rd Rough Sets Theory Workshop, pp. 1419 Milan, Italy (2011)
  • A. Elkins, A. ’Sostak, I. Uljane, On a category of extensional fuzzy rough approximation L-valued spaces, In: Communication in Computer and Information Science, vol 611 (2016). 16th. International Conference IPMU 2016, Eindhofen, The Netherlands, June 20-24, 2016, Proceedings, Part II, 36-47.
  • G. Gierz, K.H. Homan, K. Keimel, J.D. Lawson, M.W. Mislove, D.S. Scott, Continuous Lattices and Domains, Cambridge University Press, Cambridge (2003)
  • J.A. Goguen, The fuzzy Tychono theorem, J. Math. Anal. Appl. 43, 734742 (1973)
  • J.A. Goguen, L-fuzzy sets, J. Math. Anal. Appl., 18 (1967), 145174.
  • J. Hao, Q. Li, The relation between L-fuzzy rough sets and L-topology, Fuzzy Sets and Systems, 178 (2011)
  • U. Höhle, M-valued sets and sheaves over integral commutative cl-monoids Chapter 2 in Applications of Category Theory to Fuzzy Sets, 1992, Kluwer Acad. Press, S.E. Rodabaugh, E.P. Klement and U. Höhle eds, pp. 3372.
  • U. Höhle, Commutative, residuated l-monoids, in: S.E. Rodabaugh, E.P. Klement, U. Höhle eds., Non-classical logics and their applications to Fuzzy Sets. Kluwer Acad. Publ. Dodrecht, Boston 1995, pp. 53106.
  • U. Höhle, Many-valued equalities, singletons and fuzzy partitions Soft Computing 2 (1998), 134140.
  • U. Höhle, Many Valued Topology and its Application Kluwer Acad. Publ. 2001, Boston, Dodrecht, London.
  • J. Järvinen, J. Kortelainen, A unified study between modal-like operators, topologies and fuzzy sets, Fuzzy Sets and Systems 158 (2007) 12171225.
  • F. Klawonn, Fuzzy points, fuzzy relations and fuzzy functions, in: V. Novák, I. Perlieva (Eds.), Discovering the World with Fuzzy Logic, Springer, Berlin, 2000, pp. 431453.
  • E.P. Klement, R. Mesiar, E. Pap, Triangular norms, Kluwer Acad. Publ., 2000.
  • J. Kortelainen, On relationship between modified sets, topological spaces and rough sets, Fuzzy Sets and Systems 61 (1994) 91-95.
  • T. Kubiak, On fuzzy topologies, PhD Thesis, Adam Mickiewicz University Poznan, Poland (1985)
  • Liu Yingming, Luo Maokang, Fuzzy Topology Advances in Fuzzy Systems - Applications and Topology. World Scientif. Singapore, New Jersey, London, Hong Kong, 1997.
  • K. Menger, Probabilistic geometry, Proc. N.A.S. 27 (1951), 226229.
  • J.S. Mi, B.Q. Hu Topological and lattice structure of L-fuzzy rough sets determined by upper and lower sets, Information Sciences 218 (2013) 194204.
  • W. Morgan, and R. P. Dilworth Residuated lattices Trans. Amer. Math. Soc. 45 (1939) 335- 354. Reprinted in K.Bogart, R. Freese, and J. Kung eds. The Dilworth Theorems: Selected Papers of R.P. Dilworth Basel, 1990 Birkhauser.
  • Z. Pawlak, Rough sets , International J. of Computer and Information Sciences, 11 (1982) 341-356.
  • K. Qin, Z. Pei, On the topological properties of fuzzy rough sets, Fuzzy Sets and Systems 151 (2005) 601613.
  • K. Qin, Z. Pei, Generalized rough sets based on reexive and transitive relations, Information Sciences 178 (2008) 41384141.
  • A.M. Radzikowska, E.E. Kerre, A comparative study of fuzzy rough sets. Fuzzy Sets and Syst. 126, 137155 (2002)
  • G.N. Raney, A subdirect-union representation for completely distibutive complete lattices, Proc. Amer. Math. Soc. 4 (1953), 518522.
  • K.I. Rosenthal Quantales and Their Applications, Pirman Research Notes in Mathematics 234. Longman Scientic & Technical (1990)
  • B. Schweizer, A. Sklar, Statistical metric spaces, Pacic J. Math. 10, 215229.
  • A. Skowron, On the topology in information systems, Bull. Polon. Acad. Sci. Math. 36 (1988), 477480.
  • A. ’Sostak, On a fuzzy topological structure, Suppl. Rend. Circ. Mat. Palermo Ser II 11 (1985), 89-103
  • A. ’Sostak, Two decades of fuzzy topology: Basic ideas, notions and results, Russian Math. Surveys, 44 (1989), 125186
  • A. ’Sostak, Basic structures of fuzzy topology, J. Math. Sci. 78 (1996), 662701.
  • A. ’Sostak, Fuzzy functions and an extension of the category L-TOP of Chang-Goguen L- topological spaces, Proceedings of the 9th Prague Topological Symposium (2001), 271294.
  • S.P. Tiwari, A.K. Srivastava, Fuzzy rough sets. fuzzy preoders and fuzzy topoloiges, Fuzzy Sets and Syst., 210 (2013), 6368.
  • I. Ul,jane, On the order type L-valued relations on L-powersets, Soft Computing 14 (2007), 183-199.
  • L. Valverde, On the structure of F-indistinguishibility operators, Fuzzy Sets and Syst. 17 (1985) 313328.
  • A. Wiweger, On topological rough sets, Bull. Polon. Acad. Sci. Math. 37 (1988), 5162.
  • H. Yu, W.R. Zhan, On the topological properties of generalized rough sets, Information Sciences 263 (2014), 141152.
  • L. Zadeh, Fuzzy sets, Information and Control (1965)
  • L. Zadeh Similarity relations and fuzzy orderings, Inf. Sci. 3 (1971) 177200.
Year 2017, Volume: 46 Issue: 1, 15 - 32, 01.02.2017

Abstract

References

  • U. Bodenhofer, Ordering of fuzzy sets based on fuzzy orderings. I: The basic approach, Mathware Soft Comput. 15 (2008) 201218.
  • U. Bodenhofer, Ordering of fuzzy sets based on fuzzy orderings. II: Generalizations, Mathware Soft Comput. 15 (2008) 219249.
  • L.M. Brown, M. Diker, Ditopological texture spaces and intuitionistic sets Fuzzy Sets and Syst. 98, (1998) 217224.
  • L.M. Brown, R. Ertürk, “S. Dost, Ditopological texture spaces and fuzzy topology, I. Basic concepts, Fuzzy Sets and Syst. 147, (2004) 171199.
  • L.M. Brown, R. Ertürk, “S. Dost, Ditopological texture spaces and fuzzy topology, II. Topo- logical considerations, Fuzzy Sets and Syst. 147, (2004) 201231. 18861912.
  • L.M. Brown, A. ’Sostak, Categories of fuzzy topologies in the context of graded ditopologies, Iranian J. Of Fuzzy Systems, Systems 11, No. 6, (2014) pp. 1-20
  • C.L. Chang, Fuzzy topological spaces, J. Math. Anal. Appl. 24, 182190 (1968)
  • K.C. Chattopadhyay, R.N. Nasra, S.K. Samanta, Gradation of openness: Fuzzy topology, Fuzzy Sets and Syst. 48 (1992), 237242.
  • K.C. Chattopadhyay, S.K. Samanta, Fuzzy closure operators, fuzzy compactness and fuzzy connectedness, Fuzzy Sets and Syst. 54 (1992), 237242.
  • P. Chen, D. Zhang, Alexandro L-cotopological spaces, Fuzzy Sets and Systems 161 (2010) 2505  2525.
  • M. Demirci, Fuzzy functions and their fundamental properties, Fuzzy Sets Syst. 106 (1999), 239246
  • D. Dubois, H. Prade, Rough fuzzy sets and fuzzy rough sets. Internat. J. General Syst. 17, 191209 (1990)
  • A. Elkins, A. ’Sostak, On some categories of approximate systems generated by L-relations. In: 3rd Rough Sets Theory Workshop, pp. 1419 Milan, Italy (2011)
  • A. Elkins, A. ’Sostak, I. Uljane, On a category of extensional fuzzy rough approximation L-valued spaces, In: Communication in Computer and Information Science, vol 611 (2016). 16th. International Conference IPMU 2016, Eindhofen, The Netherlands, June 20-24, 2016, Proceedings, Part II, 36-47.
  • G. Gierz, K.H. Homan, K. Keimel, J.D. Lawson, M.W. Mislove, D.S. Scott, Continuous Lattices and Domains, Cambridge University Press, Cambridge (2003)
  • J.A. Goguen, The fuzzy Tychono theorem, J. Math. Anal. Appl. 43, 734742 (1973)
  • J.A. Goguen, L-fuzzy sets, J. Math. Anal. Appl., 18 (1967), 145174.
  • J. Hao, Q. Li, The relation between L-fuzzy rough sets and L-topology, Fuzzy Sets and Systems, 178 (2011)
  • U. Höhle, M-valued sets and sheaves over integral commutative cl-monoids Chapter 2 in Applications of Category Theory to Fuzzy Sets, 1992, Kluwer Acad. Press, S.E. Rodabaugh, E.P. Klement and U. Höhle eds, pp. 3372.
  • U. Höhle, Commutative, residuated l-monoids, in: S.E. Rodabaugh, E.P. Klement, U. Höhle eds., Non-classical logics and their applications to Fuzzy Sets. Kluwer Acad. Publ. Dodrecht, Boston 1995, pp. 53106.
  • U. Höhle, Many-valued equalities, singletons and fuzzy partitions Soft Computing 2 (1998), 134140.
  • U. Höhle, Many Valued Topology and its Application Kluwer Acad. Publ. 2001, Boston, Dodrecht, London.
  • J. Järvinen, J. Kortelainen, A unified study between modal-like operators, topologies and fuzzy sets, Fuzzy Sets and Systems 158 (2007) 12171225.
  • F. Klawonn, Fuzzy points, fuzzy relations and fuzzy functions, in: V. Novák, I. Perlieva (Eds.), Discovering the World with Fuzzy Logic, Springer, Berlin, 2000, pp. 431453.
  • E.P. Klement, R. Mesiar, E. Pap, Triangular norms, Kluwer Acad. Publ., 2000.
  • J. Kortelainen, On relationship between modified sets, topological spaces and rough sets, Fuzzy Sets and Systems 61 (1994) 91-95.
  • T. Kubiak, On fuzzy topologies, PhD Thesis, Adam Mickiewicz University Poznan, Poland (1985)
  • Liu Yingming, Luo Maokang, Fuzzy Topology Advances in Fuzzy Systems - Applications and Topology. World Scientif. Singapore, New Jersey, London, Hong Kong, 1997.
  • K. Menger, Probabilistic geometry, Proc. N.A.S. 27 (1951), 226229.
  • J.S. Mi, B.Q. Hu Topological and lattice structure of L-fuzzy rough sets determined by upper and lower sets, Information Sciences 218 (2013) 194204.
  • W. Morgan, and R. P. Dilworth Residuated lattices Trans. Amer. Math. Soc. 45 (1939) 335- 354. Reprinted in K.Bogart, R. Freese, and J. Kung eds. The Dilworth Theorems: Selected Papers of R.P. Dilworth Basel, 1990 Birkhauser.
  • Z. Pawlak, Rough sets , International J. of Computer and Information Sciences, 11 (1982) 341-356.
  • K. Qin, Z. Pei, On the topological properties of fuzzy rough sets, Fuzzy Sets and Systems 151 (2005) 601613.
  • K. Qin, Z. Pei, Generalized rough sets based on reexive and transitive relations, Information Sciences 178 (2008) 41384141.
  • A.M. Radzikowska, E.E. Kerre, A comparative study of fuzzy rough sets. Fuzzy Sets and Syst. 126, 137155 (2002)
  • G.N. Raney, A subdirect-union representation for completely distibutive complete lattices, Proc. Amer. Math. Soc. 4 (1953), 518522.
  • K.I. Rosenthal Quantales and Their Applications, Pirman Research Notes in Mathematics 234. Longman Scientic & Technical (1990)
  • B. Schweizer, A. Sklar, Statistical metric spaces, Pacic J. Math. 10, 215229.
  • A. Skowron, On the topology in information systems, Bull. Polon. Acad. Sci. Math. 36 (1988), 477480.
  • A. ’Sostak, On a fuzzy topological structure, Suppl. Rend. Circ. Mat. Palermo Ser II 11 (1985), 89-103
  • A. ’Sostak, Two decades of fuzzy topology: Basic ideas, notions and results, Russian Math. Surveys, 44 (1989), 125186
  • A. ’Sostak, Basic structures of fuzzy topology, J. Math. Sci. 78 (1996), 662701.
  • A. ’Sostak, Fuzzy functions and an extension of the category L-TOP of Chang-Goguen L- topological spaces, Proceedings of the 9th Prague Topological Symposium (2001), 271294.
  • S.P. Tiwari, A.K. Srivastava, Fuzzy rough sets. fuzzy preoders and fuzzy topoloiges, Fuzzy Sets and Syst., 210 (2013), 6368.
  • I. Ul,jane, On the order type L-valued relations on L-powersets, Soft Computing 14 (2007), 183-199.
  • L. Valverde, On the structure of F-indistinguishibility operators, Fuzzy Sets and Syst. 17 (1985) 313328.
  • A. Wiweger, On topological rough sets, Bull. Polon. Acad. Sci. Math. 37 (1988), 5162.
  • H. Yu, W.R. Zhan, On the topological properties of generalized rough sets, Information Sciences 263 (2014), 141152.
  • L. Zadeh, Fuzzy sets, Information and Control (1965)
  • L. Zadeh Similarity relations and fuzzy orderings, Inf. Sci. 3 (1971) 177200.
There are 50 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Alexander \v{s}ostak This is me

Aleksandrs Elkins This is me

Publication Date February 1, 2017
Published in Issue Year 2017 Volume: 46 Issue: 1

Cite

APA \v{s}ostak, A., & Elkins, A. (2017). $L^M$-valued equalities, $L^M$-rough approximation operators and $ML$-graded ditopologies. Hacettepe Journal of Mathematics and Statistics, 46(1), 15-32.
AMA \v{s}ostak A, Elkins A. $L^M$-valued equalities, $L^M$-rough approximation operators and $ML$-graded ditopologies. Hacettepe Journal of Mathematics and Statistics. February 2017;46(1):15-32.
Chicago \v{s}ostak, Alexander, and Aleksandrs Elkins. “$L^M$-Valued Equalities, $L^M$-Rough Approximation Operators and $ML$-Graded Ditopologies”. Hacettepe Journal of Mathematics and Statistics 46, no. 1 (February 2017): 15-32.
EndNote \v{s}ostak A, Elkins A (February 1, 2017) $L^M$-valued equalities, $L^M$-rough approximation operators and $ML$-graded ditopologies. Hacettepe Journal of Mathematics and Statistics 46 1 15–32.
IEEE A. \v{s}ostak and A. Elkins, “$L^M$-valued equalities, $L^M$-rough approximation operators and $ML$-graded ditopologies”, Hacettepe Journal of Mathematics and Statistics, vol. 46, no. 1, pp. 15–32, 2017.
ISNAD \v{s}ostak, Alexander - Elkins, Aleksandrs. “$L^M$-Valued Equalities, $L^M$-Rough Approximation Operators and $ML$-Graded Ditopologies”. Hacettepe Journal of Mathematics and Statistics 46/1 (February 2017), 15-32.
JAMA \v{s}ostak A, Elkins A. $L^M$-valued equalities, $L^M$-rough approximation operators and $ML$-graded ditopologies. Hacettepe Journal of Mathematics and Statistics. 2017;46:15–32.
MLA \v{s}ostak, Alexander and Aleksandrs Elkins. “$L^M$-Valued Equalities, $L^M$-Rough Approximation Operators and $ML$-Graded Ditopologies”. Hacettepe Journal of Mathematics and Statistics, vol. 46, no. 1, 2017, pp. 15-32.
Vancouver \v{s}ostak A, Elkins A. $L^M$-valued equalities, $L^M$-rough approximation operators and $ML$-graded ditopologies. Hacettepe Journal of Mathematics and Statistics. 2017;46(1):15-32.