The modularity condition for uninorms and t-operators

Uninorm solutions and (or) nullnorm solutions to the modularity condition equations

“…Notice that uninorms in CU have more properties of t-norms and t-conorms than the ones in U. Hence the results obtained in this paper are much simpler than those of [11] and similar to those of equations only involving t-norms and t-conorms.…”

“…The aim of the present paper is to investigate modularity condition equations where the uninorms involved are in CU and hence di er from the ones in [11]. Notice that uninorms in CU have more properties of t-norms and t-conorms than the ones in U.…”

“…Furthermore, t-operators were introduced in [10], which are also the generalizations of the notions of t-norms and t-conorms and can be obtained by weakening their boundary conditions. In [11], it is pointed out that nullnorms and t-operators are equivalent since the two operators have the same block structures on [0; 1] 2 . Namely, if a binary operator F is a nullnorm then F is also a t-operator and vice versa.…”

“…Hence it is interesting to study them. Symbols and concepts in this paper may be also traced back to [7,11]. In addition, uninorms, nullnorms and t-operators were investigated in [1,2,4,6,9,12,13].…”

Binary Operations on Fuzzy Sets: Recent Advances

“…Recall that classes of uninorms and t-operators satisfying some algebraic properties (such as modularity, distributivity, reversibility) as well as their structure on finite totally ordered scales were studied in [37,32,33,34].…”