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In this paper, we consider the problem of representation and construction of intuitionistic fuzzy preorders and weak orders, where many fundamental representation results extending those of Ulrich Bodenhofer et al. are presented.

In a recent paper, Çeven and Öztürk have generalized the notion of derivation on a lattice to f-derivation, where f is a given function of that lattice into itself. Under some conditions, they have characterized the distributive and modular lattices in terms of their isotone f-derivations. In this paper, we investigate the most important properties of isotone f-derivations on a lattice, paying particular attention to the lattice (resp. ideal) structures of isotone f-derivations and the sets of their f-fixed points. As applications, we provide characterizations of distributive lattices and principal ideals of a lattice in terms of principal f-derivations.

The main goal of this paper is to investigate the aggregation of diverse families of binary fuzzy relations, fuzzy filters, and fuzzy lattices. Some links between these families and their images via an aggregation are explored.

The aim of this article is to develop a representation theory of interval-valued Łukasiewicz–Moisil algebras; the concept of interval fuzzy sets involves the role that the notion of field of sets plays for the representation of Boolean algebras. This theory provides both a semantic interpretation of a Łukasiewicz interval-valued logic and a logical basis for the interval fuzzy sets theory.

In the present paper, we introduce the notion of (F,G)-derivation on a lattice as a generalization of the notion of (,)-derivation. This newly notion is based on two arbitrary binary operations F and G instead of the meet () and the join () operations. Also, we investigate properties of (F,G)-derivation on a lattice in details. Furthermore, we deĄne and study the notion of principal (F,G)-derivations as a particular class of (F,G)-derivations. As applications, we provide two representations of a given lattice in terms of its principal (F,G)-derivations. [ABSTRACT FROM AUTHOR]

In this paper, some characterizations of fuzzy filters and their prime filters are given. Also, we characterize fuzzy filters and fuzzy prime filters using their α--cuts. Finally, we give necessary and sufficient conditions under which a fuzzy set is a prime filter and we introduce the notion of fuzzy lattices isomorphism. [ABSTRACT FROM AUTHOR]

A representation theorem for bounded distributive hyperlattices is given. The equivalence between the category of Priestley spaces and the dual of the category of bounded distributive hyperlattices is established. [ABSTRACT FROM AUTHOR]