Fuzzy implications play important roles in both theoretic and applied communities of fuzzy set theory and have been extensively investigated. Fuzzy coimplications, as the dual operators of fuzzy implications, have received, however, little attention in the literature and, to the best of our knowledge, (T, N)-coimplications and residual coimplications are the only well-studied classes. On the other hand, many studies show that not all properties of fuzzy implications can be dualized in the standard way to fuzzy coimplications while fuzzy coimplications play key roles in quantum B-algebras, BCK-algebras, Heyting-Brouwer logic, fuzzy expert systems, etc. The present paper chooses to stand in the line of research on fuzzy coimplications. The notion of (O, N)-coimplication, obtained by relaxing the t-norm T in a (T, N)-coimplication to be an overlap function O that is used to model the degree to which an object is simultaneously supported by two different classes in decision making and image processing, is introduced, and basic properties and characterizations of some subclasses of (O, N)-coimplications are obtained.
AlsinaC., FrankM.J., SchweizerB.AssociativeFunctions, Triangular Norms and Copulas, World Scientific Publishing Company, Singapore2006.
2.
BaczyńskiM., Distributivity of implication operations overtrepresentable t-norms in interval-valued fuzzy set theory: Thecase of nilpotent t-norms, Information Sciences257(2014), 388–399.
BaczyńskiM. and JayaramB., On the characterization of (S, N)-implications, Fuzzy Sets and Systems158(15) (2007), 1713–1727.
5.
BaczyńskiM. and JayaramB., (S, N)- and R-implications: A state-of-the-art survey, Fuzzy Sets and Systems159(14) (2008), 1836–1859.
6.
BaczyńskiM. and JayaramB., (U, N)-implications and their characterizations, Fuzzy Sets and Systems160(14) (2009), 2049–2062.
7.
BedregalB.C., On interval fuzzy negations, Fuzzy Sets andSystems161(17) (2010), 2290–2313.
8.
BedregalB., DimuroG.P., BustinceH. and BarrenecheaE., Newresults on overlap and grouping functions, Information Sciences249 (2013), 148–170.
9.
BedregalB.C., DimuroG.P., ReiserR.H.S.An approach to interval-valued R-implications and automorphisms, in: CarvalhoJ.P., DuboisD., KaymakU., da Costa SousaJ.M. (Eds.). Proceedings of the Joint 2009 International Fuzzy Systems Association World Congress and 2009 European Society of Fuzzy Logic and Technology Conference, IFSA/EUSFLAT, 2009, pp. 1–6.
10.
BedregalB.C., DimuroG.P., SantiagoR.H.N. and ReiserR.H.S., On interval fuzzy S-implications, Information Sciences180(8) (2010), 1373–1389.
11.
BedregalB. and SantiagoR.H.N., Interval representations, Łukasiewicz implicators and Smets-Magrez axioms, Information Sciences221 (2013), 192–200.
12.
BeliakovG., PraderaA., CalvoT.Aggregation Functions: A Guide for Practitioners, Springer, Berlin, 221, 2007.
13.
BustinceH., BarrenecheaE. and PagolaM., Image thresholding using restricted equivalence functions and maximizing the measuresof similarity, Fuzzy Sets and Systems158(5) (2007), 496–516.
14.
BustinceH., BurilloP. and SoriaF., Automorphism, negations andimplication operators, Fuzzy Sets and Systems134(2) (2003), 209–229.
BustinceH., FernandezJ., MesiarR., MonteroJ. and OrdunaOverlapR., Overlap index, overlap functions and migrativity, in: Proceedings of IFSA/EUSFLAT Conference, 2009, pp. 300–305.
17.
BustinceH., PagolaM., MesiarR., HullermeierE. and HerreraF., Grouping overlap, and generalized bientropic functions for fuzzymodeling of pairwise comparisons, IEEE Transactions on Fuzzy Systems20(3) (2012), 405–415.
18.
CintulaP., HájekP., NogueraC.Handbook of Mathematical Fuzzy Logic, London: College Publications, 1, 2011.
19.
CintulaP., HájekP., NogueraC.Handbook of Mathematical Fuzzy Logic, London: College Publications, 2, 2011.
20.
De BaetsB., Coimplicators, the forgotten connectives, Tatra Mountains Mathematical Publications12 (1997), 229–240.
21.
De BaetsB. and FodorJ.C., Residual operators of uninorms, Soft Computing3(2) (1999), 88–100.
22.
DimuroG.P., On interval fuzzy numbers, in 2011 Workshop School on Theoretical Computer Science WEIT 2011, Los Alamitos: IEEE, 2011, pp. 3–8.
23.
DimuroG.P. and BedregalB., On additive generators of groupingfunctions, Springer International Publishing444(2014), 252–261.
24.
DimuroG.P. and BedregalB., On (G,N)-implications derived fromgrouping functions, Information Sciences279 (2014), 1–17.
25.
DimuroG.P. and BedregalB., On residual implications derived fromoverlap functions, Information Sciences312 (2015), 78–88.
26.
DimuroG.P. and BedregalB., On the laws of contraposition for residual implications derived from overlap functions, in: 2015 IEEE International Conference on Fuzzy Systems (FUZZIEEE) IEEE, Los Alamitos, 2015, pp. 1–7.
27.
DimuroG.P. and BedregalB., Archimedean overlap functions: Theordinal sum and the cancellation, idempotency and limitingproperties, Fuzzy Sets and Systems252 (2014), 39–54.
28.
DimuroG.P., BedregalB., BustinceH., AsiainM.J. and MesiarR., On additive generators of overlap functions, Fuzzy Sets and Systems287 (2016), 76–96.
29.
DimuroG.P., BedregalB.C., SantiagoR.H.N. and ReiserR.H.S., Interval additive generators of interval t-norms and intervalt-conorms, Information Sciences181(18) (2011), 3898–3916.
30.
DuranteF., KlementE., MesiarR. and SempiC., Conjunctors and their residual implicators: Characterizations and constructionmethods, Mediterranean Journal of Mathematics4(3) (2007), 343–356.
31.
ElkanoM., GalarM., SanzJ. and BustinceH., Fuzzy rulebased classification systems for multi-class problems using binary decomposition strategies: On the influence of n-dimeusional overlap functions in the fuzzy reasoning method, Information Sciences332 (2016), 94–114.
32.
ElkanoM., GalarM., SanzJ., FernándezA., BarrenecheaE.HerreraF. and BustinceH., Enhancing multi-class classification in FARC-HD fuzzy classifier: On the synergy between ndimensional overlap functions and decomposition strategies, IEEE Transactions on Fuzzy Systems23(5) (2015), 1562–1580.
GottwaldS., A Treatise on Many-Valued Logic. Resarch Studies Press, Baldok, 2001.
35.
JebrilI.H., (T, N) and residual fuzzy co-implication in dualHeyting algebra with applications, Licensee Preprints (2016).
36.
JurioA., BustinceH., PagolaM., PraderaA. and YagerR., Some properties of overlap and grouping functions and their applicationto image thresholding, Fuzzy Sets and Systems229(2013), 69–90.
MasM., MonserratM. and TorrensJ., QL-implications versusD-implications, Kybernetika42(3) (2006), 351–366.
39.
MesiarR. and MesiarováA., Residual implications andleftcontinuous t-norms which are ordinal sums of igroups, Fuzzy Sets and Systems143(1) (2004), 47–57sem.
40.
OhK.-W. and KandelA., Coimplication and its applications to fuzzy expert systems, Information Sciences56(1-3) (1991), 59–73.
41.
ReiserR.H., BedregalB. and BaczyńskiM., Aggregating fuzzy implications, Information Sciences253 (2013), 126–146.
42.
ReiserR.H.S., DiasM., BedregalB. and SantiagoR.H.N., Canonical representation of the Yager’s classes of fuzzy implications, Computational & Applied Mathematics32(3) (2013), 401–412.
43.
RumpW., Quantum B-algebra, Central European Journal ofMathematics11(11) (2013), 1881–1899.
44.
RumpW. and YangY.C., Non-commutative logical algebras and algebraic quantales, Annals of Pure and Applied Logic165(2) (2014), 759–785.
45.
SuY. and LiuH.W., Characterizations of residual coimplications of pseudo-uninorms on a complete lattice, Fuzzy Sets and Systems261 (2015), 44–59.
46.
SuY. and WangZ.D., Constructing implications and coimplicationson a complete lattice, Fuzzy Sets and Systems247(2014), 68–80.
47.
SuY., XieA.F. and LiuH.W., On ordinal sum implications, Information Sciences293 (2015), 251–262.
48.
SuY., ZongW.W. and LiuH.W., Distributivity of the ordinal sum implications over t-norms and t-conorms, IEEE Transactions onFuzzy Systems24(4) (2016), 827–840.
49.
TripathyB.C., BaruahA., EtM. and GungorM., On almost statistical convergence of new type of generalized difference sequence of fuzzy numbers, Iranian Journal of Science and Technology, Transactions A: Science36(2) (2012), 147–155.
50.
TripathyB.C. and GoswamiR., Multiple sequences in probabilistic normed spaces, Afrika Matematika26(5-6) (2015), 753–760.
51.
TripathyB.C. and GoswamiR., On triple difference sequences ofreal numbers in probabilistic normed spaces, Proyecciones Journal of Mathematics33(2) (2014), 157–174.
52.
WolterF., On logics with coimplication, Journal of Philosophical Logic27(4) (1998), 353–387.
53.
XieA.F. and LiuH.W., A generalization of Yager’s f-generated implications, International Journal of Approximate Reasoning54(1) (2013), 35–46.
54.
YagerR.R., On some new classes of implication operators and theirrole in approximate reasoning, Information Sciences167(1-4) (2004), 193–216.
55.
ZhangD.X., Triangular norms on partially ordered sets, Fuzzy Sets and Systems153(2) (2005), 195–209.
56.
ZhangF.X. and LiuH.W., On a new class of implications: (g, u)-implications and the distributive equations, International Journal of Approximate Reasoning54(8) (2013), 1049–1065.
57.
ZhouH.J.Probabilistically Quantitative Logic and its Applications, Science Press, Beijing, 2015.
