GENERALIZED RESIDUATED LATTICES BASED F-TRANSFORM

Document Type: Research Paper

Authors

1 Department of Applied Mathematics, Indian Institute of Technology (ISM), Dhanbad-826004, Jharkhand, India

2 University of Ostrava, Institute for Research and Applications of Fuzzy Modeling, NSC IT4Innovations, 30. dubna 22, 701 03 Ostrava 1, Czech Republic

Abstract

The aim of the present work is to study the  $F$-transform over a generalized residuated lattice.  We discuss the properties that are common with the $F$-transform over a residuated lattice. We show that the $F^{\uparrow}$-transform can be used in establishing a fuzzy (pre)order on the set of fuzzy sets.

Keywords


[1] A. A. Abdel-Hamid and N. N. Morsi, Associatively tied implications, Fuzzy Sets and Systems,
136(3) (2003), 291-311.
[2] K. Blount and C. Tsinakis, The structure of residuated lattices, International Journal of
Algebra and Computation, 13(4) (2003), 437-461.
[3] D. Boixader and J. Jacas, Extensionality based approximate reasoning, International Journal
of Approximate Reasoning, 19(3-4) (1998), 221-230.
[4] C. Cornelis, J. Medina, N. Verbiest, Multi-adjoint fuzzy rough sets: Defi nition, properties and
attribute selection, International Journal of Approximate Reasoning, 55(1)(2014), 412-426.
[5] J. C. Diaz and J. Medina, Multi-adjoint relation equations: Defi nition, properties and solu-
tions using concept lattices, Information Sciences, 253 (2013), 100-109.
[6] P. Flondor, G. Georgescu and A. Iorgulescu, Pseudo-t-norms and pseudo-BL algebras, Soft
Computing, 5(5) (2001), 355-371.
[7] N. Galatos, P. Jipsen, T. Kowalski and H. Ono, Residuated Lattices. An Algebraic Glimpse
at Substructural Logics, Elsevier, 2007.
[8] P. Hajek, Metamathematics of Fuzzy Logic, Kluwer Academic Publishers, Boston, 1998.
[9] P. Hajek, Observations on non-commutative fuzzy logic, Soft Computing, 8(1) (2003), 38-43.
[10] N. T. Hung and E. A. Walker, A First Course in Fuzzy Logic, CRC Press, Boca Raton, 1997.
[11] A. Khastan, I. Perfi lieva and Z. Alizani, A new fuzzy approximation method to Cauchy
problems by fuzzy transform, Fuzzy Sets and Systems, 288 (2016), 75-95.
[12] M. Liu, D. Chen, C. Wu and H. Li, Approximation theorem of the fuzzy transform in fuzzy
reasoning and its application to the scheduling problem, Computers & Mathematics with
Applications, 51(3-4) (2006), 515-526.
[13] F. D. Martino, V. Loia and S. Sessa, Fuzzy transforms method in prediction data analysis,
Fuzzy Sets and Systems, 180(1) (2011), 146-163.
[14] J. Medina, M. Ojeda-Aciego and J. Ruiz-Calvino, Formal concept analysis via multi-adjoint
concept lattices, Fuzzy Sets and Systems, 160(2) (2009), 130-144.

[15] J. Medina, M. Ojeda-Aciego, A. Valverde and P. Vojtas, Towards biresiduated multi-adjoint
logic programming, Lecture Notes in Arti ficial Intelligence, 3040 (2004), 608-617.
[16] V. Novak, I. Perfi lieva, M. Holcapek and V. Kreinovich, Filtering out high frequencies in
time series using F-transform, Information Sciences, 274 (2014), 192-209.
[17] I. Perfi lieva, Fuzzy transforms: Theory and applications, Fuzzy Sets and Systems, 157(8)
(2006), 993-1023.
[18] I. Perfi lieva, Fuzzy transforms and their applications to image compression, Lecture Notes in
Computer Science, 3849 (2006), 19-31.
[19] I. Perfi lieva, Fuzzy transforms: A challenge to conventional transforms, Advances in Image
and Electron Physics, 147 (2007), 137-196.
[20] I. Per filieva, Finitary solvability conditions for systems of fuzzy relation equations, Informa-
tion Sciences, 234 (2013), 29-43.
[21] I. Perfi lieva, D. Dubois, H. Prade, F. Esteva, L. Godo and P. Hodakova, interpolation of
fuzzy data. analytical approach and overview, Fuzzy Sets and Systems, 192 (2012), 134-158.
[22] I. Perfi lieva, M. Holcapek and V. Kreinovich, A new reconstruction from the F-transform
components, Fuzzy Sets and Systems, 288 (2016), 3-25.
[23] I. Perfi lieva, V. Novak and A. Dvorak, Fuzzy transforms in the analysis of data, International
Journal of Approximate Reasoning, 48(1) (2008), 36-46.
[24] I. Perfi lieva, A. P. Singh and S. P. Tiwari, On the relationship among F-transform, fuzzy
rough set and fuzzy topology, Soft Computing, 21(13) (2017), 3513-3521.
[25] I. Perfi lieva and R. Valasek, Fuzzy transforms in removing noise, Advances in Soft Comput-
ing, 2 (2005), 221-230.
[26] E. H. Ruspini, On the semantics of fuzzy logic, International Journal of Approximate Rea-
soning, 5(1) (1993), 45-88.
[27] C. Russo, Quantale Modules with Applications to Logic and Image Processing, Lambert
Academic Publishers, Saarbrucken, 2009.
[28] C. Russo, Quantale modules and their operators, with applications, Journal of Logic and
Computation, 20(4) (2010), 917-946.
[29] A. P. Singh and S. P. Tiwari, F-transform based on generalized residuated lattices, Decision
Making and Soft Computing, 9 (2014), 720-725.
[30] L. Stefanini, F-transform with parametric generalized fuzzy partitions, Fuzzy Sets and Sys-
tems, 180(1) (2011), 98-120.
[31] M. Stepnicka and O. Polakovic, A neural network approach to the fuzzy transform, Fuzzy
Sets and Systems, 160(8) (2009), 1037-1047.
[32] O. Strauss, Non-additive interval-valued F-transform, Fuzzy Sets and Systems, 270 (2014),
1-24.
[33] L. Troiano and P. Kriplani, Supporting trading strategies by inverse fuzzy transform, Fuzzy
Sets and Systems, 180(1) (2011), 121-145.
[34] C. Y. Wang and B. Q. Hu, Fuzzy rough sets based on generalized residuated lattices, Infor-
mation Sciences, 248 (2013), 31-49.