In this paper, we generally study an order induced by nullnorms on bounded lattices. We investigate monotonicity property of nullnorms on bounded lattices with respect to the F-partial order. Also, we introduce the set of incomparable elements with respect to the F-partial order for any nullnorm on a bounded lattice. Finally, we investigate the relationship between the order induced by a nullnorm and the distributivity property for nullnorms.
A meet semilattice with a partial join operation satisfying certain axioms is a JP-semilattice. A PJP-semilattice is a pseudocomplemented JP-semilattice. In this paper we describe the smallest PJP-congruence containing a kernel ideal as a class. Also we describe the largest PJP-congruence containing a filter as a class. Then we give several characterizations of congruence kernels and cokernels for distributive PJP-semilattices.
We present a formal scheme which whenever satisfied by relations of a given relational lattice L containing only reflexive and transitive relations ensures distributivity of L.
In 2015, a new class of fuzzy implications, called ordinal sum implications, was proposed by Su et al. They then discussed the distributivity of such ordinal sum implications with respect to t-norms and t-conorms. In this paper, we continue the study of distributivity of such ordinal sum implications over two newly-born classes of aggregation operators, namely overlap and grouping functions, respectively. The main results of this paper are characterizations of the overlap and/or grouping function solutions to the four usual distributive equations of ordinal sum fuzzy implications. And then sufficient and necessary conditions for ordinal sum implications distributing over overlap and grouping functions are given.
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.
Distributivity of fuzzy implications over different fuzzy logic connectives have a very important role to play in efficient inferencing in approximate reasoning, especially in fuzzy control systems (see \cite{Combs_Andrews_1998,Jayaram2008} and \cite{Baczynski_Jayaram_2008}). Recently in some considerations connected with these distributivity laws, the following functional equation appeared (see \cite{Baczynski_Jayaram_2009})
f(min(x+y,a))=min(f(x)+f(y),b),
where a,b>0 and f:[0,a]→[0,b] is an unknown function. In this paper we consider in detail a generalized version of this equation, namely the equation
f(m1(x+y))=m2(f(x)+f(y)),
where m1,m2 are functions defined on some intervals of \R satisfying additional assumptions. We analyze the cases when m2 is injective and when m2 is not injective.