A large number of studies have investigated the systems of fuzzy relation equations and inequalities, which have a much wider field of application. In this paper, we study several types of systems of fuzzy relation equations and inequalities consisting of a given family of k-ary fuzzy relations, where natural number k ≥ 2, and three unknown fuzzy relations over complete residuated lattices and meet-continuous lattices. Their solutions are triples of fuzzy relations. For the systems of fuzzy relation inequalities, we give the greatest solutions contained in a given triple of fuzzy relations, the least solutions containing a given triple of fuzzy relations, or give maximal solutions contained in or containing a given triple of fuzzy relations, or belonging to a given interval of triples of fuzzy relations over complete residuated lattices and complete meet-continuous lattices. For the systems of fuzzy relation equations, we present a method of computing maximal solutions contained in a given triple of fuzzy relations and a method of computing minimal solutions containing a given triple of fuzzy relations. Furthermore, we provide some conditions under which there exist the greatest solutions contained in and the least solutions containing a given triple of fuzzy relations.
BělohlávekR., Fuzzy Relational Systems: Foundations and Principles, Kluwer, New York, 2002.
2.
BělohlávekR. and VychodilV., Fuzzy Equational Logic, Studies in Fuzziness and Soft Computing, Springer, Berlin-Heidelberg, 2005.
3.
BatageljV., Large scale social network analysis, in: R.A. Meyers (Ed.), Encyclopedia of Complexity and Systems Science, Springer, New York, 2009, pp. 8245–8265.
4.
BorgattiS.P., Two-mode concepts in social network analysis, in: R.A. Meyers (Ed.), Encyclopedia of Complexity and Systems Science, Springer,NewYork, 2009, pp. 8279–8291.
5.
BartlE., Minimal solutions of generalized fuzzy relational equations: Probabilistic algorithm based on greedy approach, Fuzzy Sets and Systems260 (2015), 25–42.
6.
ĆirićM., StamenkovićA., IgnjatovićJ. and PetkovićT., Fuzzy relation equations and reduction of fuzzy automata, Journal of Computer and System Science76 (2010), 609–633.
7.
ĆirićM., IgnjatovićJ. and BogdanovićS., Uniform fuzzy relations and fuzzy functions, Fuzzy Sets and Systems160 (2009), 1054–1081.
8.
ĆirićM., StamenkovićA., IgnjatovićJ. and PetkovićT., Factorization of fuzzy automata, in: E. Csuhaj-Varju, Z. Ésik (Eds.), FCT2007, Lecture Notes in Computer Science, 4639, Springer, Heidelberg, 2007, 213–225.
9.
ĆirićM., IgnjatovićJ., DamljanovićN. and BašićM., Bisimulations for fuzzy automata, Fuzzy Sets and Systems 186 (2012), 100–139.
10.
ĆirićM., IgnjatovićJ., JančićI. and DamljanovićN., Computation of the greatest simulations and bisimulations between fuzzy automata, Fuzzy Sets and Systems208 (2012), 22–42.
11.
ĆirićM., IgnjatovićJ., BašićM. and JančićI., Nondeterministic automata: Simulation, bisimulation and structural equivalence, Information Sciences261 (2014), 185–218.
12.
ĆirićM., IgnjatovićJ. and StankovićI., Regular fuzzy equivalences on multi-mode multi-relational fuzzy networks, 16th world ongress of the International Fuzzy Systems Association (IFSA), 9th Conference of the European Society for fuzy Logic and Technology (EUSFLAT), Atlantis Press, (2015), 398–403.
13.
CechlárováK., Unique solvability of max-min fuzzy equations and strong regularity of matrices over fuzzy algebra, Fuzzy Sets and Systems75 (1995), 165–177.
14.
DoreianP., Positional analysis and blockmodeling, in: R.A. Meyers (Ed.), Encyclopedia of Complexity and Systems Science, Springer, New York, 2009, pp. 6913–6927.
15.
DoreianP., BatageljV. and FerligojA., Generalized Blockmodeling, Cambridge University Press, 2005.
16.
De BaetsB., Analytical solution methods for fuzzy relational equations, in: D. Dubois, H. Prade (Eds.), Fundamentals of Fuzzy Sets, Kluwer, Norwell, MA, 2000, pp. 291–340.
17.
Di NolaA., SanchezE., PedryczW. and SessaS., Fuzzy relation equations and their application to knowledge engineering, Kluwer Academic Press, Dordrecht, 1989.
18.
DuboisD. and PradeH., Fuzzy sets and Systems: Theory and Applications, Academic Press, New York, 1980.
19.
DuboisD. and PradeH., (Eds.), Fundamentals of Fuzzy Sets, The Handbooks of Fuzzy Sets Series, vol. 7, Kluwer Academic Publishers, 2000.
20.
DuY.B. and ZhuP., Collective relations of fuzzy relational structure, Journal of Intelligent and Fuzzy Systems31 (2018), 2807–2816.
21.
FanT.F., Rough set analysis of relational structures, Information Sciences221 (2013), 230–244.
22.
FanT.F., LiauC.J. and LinT.S., A theoretical investigation of regular equivalences for fuzzy graphs, International Journal of Approximate Reasoning49 (2008), 678–688.
23.
FanT.F. and LiauC.J., Logical characterizations of regular equivalence in weighted social networks, Artificial Intelligence214 (2014), 66–88.
24.
GondranM. and MinouxM., Dioïds and semirings: Links to fuzzy sets and other applications, Fuzzy Sets and Systems158 (12) (2007), 1273–1294.
25.
GuptaM.M., SaridisG.N. and GainesB.R., Fuzzy Automata and Decision Processes, North-Holland, New York, 1977.
26.
HájekP., Metamathematics of Fuzzy Logics, Kluwer Academic Publishers, Boston, 1998.
27.
IgnjatovićJ., ĆirićM. and SimovićV., Fuzzy relation equations and subsystems of fuzzy transition systems, Knowledge-Based Systems38 (2013), 48–61.
28.
IgnjatovićJ. and ĆirićM., Weakly linear systems of fuzzy relation inequalities and their applications: A brief survey, Filomat26(2) (2012), 207–241.
29.
IgnjatovićJ., ĆirićM. and BogdanovićS., On the greatest solutions to weakly linear systems of fuzzy relation inequalities and equations, Fuzzy Sets and Systems161 (2010), 3081–3113.
30.
IgnjatovićJ., ĆirićM., DamljanovićN. and JančićI., Weakly linear systems of fuzzy relation inequalities: The heterogeneous case, Fuzzy Sets and Systems199 (2012), 64–91.
31.
IgnjatovićJ., ĆirićM., ŠešeljaB. and TapavcěvićA., Fuzzy relation inequalities and equations, fuzzy quasi-orders, and closures and openings of fuzzy sets, Fuzzy Sets and Systems260 (2015), 1–24.
32.
JiménezJ., MontesS., ŠešeljaB. and TapavčevićA., Latticevalued approach to closed sets under fuzzy relations: Theory and applications, Computers and Mathematics with Applications62 (2011), 3729–3740.
33.
JiménezJ., MontesS., ŠešeljaB. and TapavčevićA., Fuzzy correspondence inequations and equations, Fuzzy Sets and Systems239 (2014), 81–90.
34.
KuratowskiC., Une méthode d’élimination des nombres transfinis des raisonnements mathématiques, Fundamenta Mathematicae3(1) (1922), 76–108.
35.
KlawonnF., Fuzzy points, fuzzy relations and fuzzy functions, in: V. Novǎk, I. Perfilieva (Eds.), Discovering World with Fuzzy Logic, Physica-Verlag, Heidelberg, 2000, 431–453.
36.
KlirG.J. and YuanB., Sets and Fuzzy Logic, Theory and Application, Prentice Hall, Englewood Cliffs, NJ, 1995.
37.
LiJ.X., The smallest solution of max-min fuzzy equations, Fuzzy Sets and Systems41(3) (1991), 317–327.
38.
LiJ.X., A new algorithm for the greatest solution of fuzzy bilinear equations, Fuzzy Sets and Systems46 (1992), 193–210.
39.
LiY.M. and WangX.P., Necessary and sufficient conditions for existence of maximal solutions for inf-Îś composite fuzzy relational equations, Computers and Mathematics with Application55 (2008), 1961–1973.
40.
LaiH. and ZhangD., Fuzzy preorder and fuzzy topology, Fuzzy Sets and Systems157 (2006), 1865–1885.
41.
MiyakoshiM. and ShimboM., Lower solutions of systems of fuzzy equations, Fuzzy Sets and Systems19 (1986), 37–46.
42.
PedryczW. and GomideF., Fuzzy Systems Engineering: Toward Human-Centric Computing, Wiley-IEEE Press, 2007.
43.
PeevaK. and KyosevY., Fuzzy Relational Calculus: Theory, Applications, and Software (with CD-ROM), Advances in Fuzzy Systems-Applications and Theory, vol. 22, World Scientific, 2004.
44.
PeevaK. and KyosevY., Algorithm for solving max-product fuzzy relational equations, Soft Computing11 (2007), 593–605.
45.
PerfilievaI., Fuzzy function as an approximate solution to a system of fuzzy relation equations, Fuzzy Sets and Systems147 (2004), 363–383.
46.
PerfilievaI. and GottwaldN., Fuzzy function as a solution to a system of fuzzy relation equations, International Journal of General Systems32 (2003), 361–372.
47.
PerfilievaI. and NovákV., System of fuzzy relation equations as a continuous model of IF-THEN rules, Information Sciences177 (2007), 3218–3227.
48.
PerfilievaI., Finitary solvability conditions for systems of fuzzy relation equations, Information Sciences234 (2013), 29–43.
49.
StankovićI., ĆirićM. and IgnjatovićJ., Fuzzy relation equations and inequalities with two unknowns and their applications, Fuzzy Sets and Systems322 (2017), 86–105.
50.
SanchezE., Equations de Relation Floues, Thèse de Doctorat, Faculté de Médecine de Marseille, 1974.
51.
SanchezE., Resolution of composite fuzzy relation equations, Information and control30 (1976), 38–48.
52.
SanchezE., Solutions in composite fuzzy relation equations: Application to medical diagnosis in Brouwerian logic, in: M.M. Gupta, G.N. Saridis, B.R. Gaines (Eds.), Fuzzy Automata and Decision Processes, North-Holland, Anmsterdam, (1997), 221–234.
53.
SanchezE., Resolution of eigen fuzzy sets equations, Fuzzy Sets and Systems1 (1978), 69–74.
54.
StamenkovicA., ĆirićM. and IgnjatovićJ., Reduction of fuzzy automata by means of fuzzy quasi-orders, Information Sciences275 (2014), 168–198.
55.
StepanovićV., Fuzzy set inequalities and equations with a meet-continuous codomain lattice, Journal of Intelligent and Fuzzy Systems34 (2018), 4009–4021.
56.
TepavčevićA., Diagonal relation as a continuous element in a weak congurence lattice, Olomouc, Proc of the International Conference General Algebra and Ordered Sets (1994), 156–163.
57.
ZornM., A remark on method in transfinite algebra, Bulletin of the American Mathematical Society41 (1935), 667–670.
58.
ŽibernaA., Blockmodeling of multilevel networks, Social networks39 (2014), 46–61.
