This paper deals with implications defined from disjunctive uninorms U by the expression I(x,y)=U(N(x),y) where N is a strong negation. The main goal is to solve the functional equation derived from the distributivity condition of these implications over conjunctive and disjunctive uninorms. Special cases are considered when the conjunctive and disjunctive uninorm are a t-norm or a t-conorm respectively. The obtained results show a lot of new solutions generalyzing those obtained in previous works when the implications are derived from t-conorms.
In this paper further development of Chebyshev type inequalities for Sugeno integrals based on an aggregation function H and a scale transformation φ is given. Consequences for T-(S-)evaluators are established.
Recently, the topic of construction methods for triangular norms (triangular conorms), uninorms, nullnorms, etc. has been studied widely. In this paper, we propose construction methods for triangular norms (t-norms) and triangular conorms (t-conorms) on bounded lattices by using interior and closure operators, respectively. Thus, we obtain some proposed methods given by Ertuğrul, Karaçal, Mesiar [15] and Çaylı [8] as results. Also, we give some illustrative examples. Finally, we conclude that the introduced construction methods can not be generalized by induction to a modified ordinal sum for t-norms and t-conorms on bounded lattices.
Recently, the topic related to the construction of triangular norms and triangular conorms on bounded lattices using ordinal sums has been extensively studied. In this paper, we introduce a new ordinal sum construction of triangular norms and triangular conorms on an appropriate bounded lattice. Also, we give some illustrative examples for clarity. Then, we show that a new construction method can be generalized by induction to a modified ordinal sum for triangular norms and triangular conorms on an appropriate bounded lattice, respectively. And we provide some illustrative examples.
This paper deals with two kinds of fuzzy implications: QL and Dishkant implications. That is, those defined through the expressions I(x,y)=S(N(x),T(x,y)) and I(x,y)=S(T(N(x),N(y)),y) respectively, where T is a t-norm, S is a t-conorm and N is a strong negation. Special attention is due to the relation between both kinds of implications. In the continuous case, the study of these implications is focused in some of their properties (mainly the contrapositive symmetry and the exchange principle). Finally, the case of non continuous t-norms or non continuous t-conorms is studied, deriving new implications of both kinds and showing that they remain strongly connected.
In this study, we introduce new methods for constructing t-norms and t-conorms on a bound\-ed lattice L based on a priori given t-norm acting on [a,1] and t-conorm acting on [0,a] for an arbitrary element a∈L∖{0,1}. We provide an illustrative example to show that our construction methods differ from the known approaches and investigate the relationship between them. Furthermore, these methods are generalized by iteration to an ordinal sum construction for t-norms and t-conorms on a bounded lattice.