Constructing uninorms via closure operators on a bounded lattice

Abstract In this paper, via closure (resp. interior) operators on a bounded lattice L, we propose a rather effective method to construct uninorms on L with a given triangular norm T (resp. triangular conorm S) on the subinterval [ 0 , e ] (resp. [ e , 1 ] ) of L. Our method encompasses as special cases the construction methods introduced by F. Karacal and R. Mesiar (2015) [16] as well as by G.D. Cayli et al. (2016) [5] . We also exploit a pair of novel closure and interior operators on complete lattices and utilize this pair of operators to construct uninorms. Several interesting examples are included to illustrate our method.

[1]  Berthold Schweizer,et al.  Probabilistic Metric Spaces , 2011 .

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

[3]  Gül Deniz Çayli,et al.  On a new class of t-norms and t-conorms on bounded lattices , 2018, Fuzzy Sets Syst..

[4]  Radko Mesiar,et al.  Nullnorms on bounded lattices , 2015, Inf. Sci..

[5]  Lluis Godo,et al.  Monoidal t-norm based logic: towards a logic for left-continuous t-norms , 2001, Fuzzy Sets Syst..

[6]  S. Weber.  ⊥-Decomposable measures and integrals for Archimedean t-conorms ⊥ , 1984 .

[7]  C. J. Everett.  Closure operators and Galois theory in lattices , 1944 .

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

[9]  Radko Mesiar,et al.  Medians and nullnorms on bounded lattices , 2016, Fuzzy Sets Syst..

[10]  Susanne Saminger,et al.  On ordinal sums of triangular norms on bounded lattices , 2006 .

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

[12]  Dexue Zhang,et al.  Triangular norms on partially ordered sets , 2005, Fuzzy Sets Syst..

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

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

[15]  Funda Karaçal,et al.  Construction of uninorms on bounded lattices , 2017, Kybernetika.

[16]  Radko Mesiar,et al.  On extensions of triangular norms on bounded lattices , 2008 .

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

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

[19]  Costas A. Drossos,et al.  Generalized t-norm structures , 1999, Fuzzy Sets Syst..