A characterization of the classes Umin and Umax of uninorms on a bounded lattice

Abstract As generalizations of the classes Umin and Umax of uninorms on the real unit interval, we introduce the classes Umin and Umax of uninorms on a bounded lattice and lay bare the structure of their members. It is shown that uninorms in Umin (resp. Umax) are characterized by a triangular conorm (resp. triangular norm) and a triangular subnorm (resp. triangular superconorm). The characterization theorems unify all the existing construction methods for these two classes of uninorms known in the literature. We also introduce four new related classes of uninorms on a bounded lattice and present dedicated construction methods using triangular subnorms and triangular superconorms. Finally, connections among the introduced classes of uninorms are discussed.

[1]  Gert de Cooman,et al.  Order norms on bounded partially ordered sets. , 1994 .

[2]  Bao Qing Hu,et al.  A new structure for uninorms on bounded lattices , 2020, Fuzzy Sets Syst..

[3]  Radko Mesiar,et al.  Uninorms on bounded lattices , 2015, Fuzzy Sets Syst..

[4]  Bao Qing Hu,et al.  New constructions of uninorms on bounded lattices , 2019, Int. J. Approx. Reason..

[5]  Radko Mesiar,et al.  On the construction of uninorms on bounded lattices , 2020, Fuzzy Sets Syst..

[6]  Yao Ouyang,et al.  Constructing uninorms via closure operators on a bounded lattice , 2020, Fuzzy Sets Syst..

[7]  Sanjiang Li,et al.  On constructing the largest and smallest uninorms on bounded lattices , 2020, Fuzzy Sets Syst..

[8]  Gül Deniz Çayli,et al.  New methods to construct uninorms on bounded lattices , 2019, Int. J. Approx. Reason..

[9]  Gül Deniz Çayli,et al.  A characterization of uninorms on bounded lattices by means of triangular norms and triangular conorms , 2018, Int. J. Gen. Syst..

[10]  Gül Deniz Çayli,et al.  Alternative approaches for generating uninorms on bounded lattices , 2019, Inf. Sci..

[11]  Radko Mesiar,et al.  Triangular norms on product lattices , 1999, Fuzzy Sets Syst..

[12]  Martin Kalina,et al.  Construction of uninorms on bounded lattices , 2014, 2014 IEEE 12th International Symposium on Intelligent Systems and Informatics (SISY).

[13]  Andrea Mesiarov-Zemnkov,et al.  Characterization of uninorms with continuous underlying t-norm and t-conorm by means of the ordinal sum construction , 2017 .

[14]  Joan Torrens,et al.  A survey on the existing classes of uninorms , 2015, J. Intell. Fuzzy Syst..

[15]  Ronald R. Yager,et al.  Uninorm aggregation operators , 1996, Fuzzy Sets Syst..

[16]  Bernard De Baets,et al.  Idempotent uninorms , 1999, Eur. J. Oper. Res..

[17]  Benjamín R. C. Bedregal,et al.  Extension of fuzzy logic operators defined on bounded lattices via retractions , 2012, Comput. Math. Appl..

[18]  Radko Mesiar,et al.  Characterization of uninorms on bounded lattices , 2017, Fuzzy Sets Syst..

[19]  Gül Deniz Çayli,et al.  On the structure of uninorms on bounded lattices , 2019, Fuzzy Sets Syst..

[20]  A. H. Clifford,et al.  Naturally Totally Ordered Commutative Semigroups , 1954 .

[21]  Wei Ji.  Constructions of uninorms on bounded lattices by means of t-subnorms and t-subconorms , 2021, Fuzzy Sets Syst..

[22]  Ronald R. Yager,et al.  Structure of Uninorms , 1997, Int. J. Uncertain. Fuzziness Knowl. Based Syst..

[23]  Radko Mesiar,et al.  On a new class of uninorms on bounded lattices , 2016, Inf. Sci..

[24]  Gül Deniz Çaylı,et al.  Uninorms on bounded lattices with the underlying t-norms and t-conorms , 2020, Fuzzy Sets Syst..