Fuzzy Ordering Relation and Fuzzy Poset

In this paper we consider the notion of Fuzzy Lattices, which was introduced by Chon (Korean J. Math 17 (2009), No. 4, 361-374). We propose some new notions for Fuzzy Ideals and Filters and provide a characterization of Fuzzy Ideals via α-level Sets and Support. Some types of ideals and filters, such as: Fuzzy Principal Ideals (Filters), Proper Fuzzy Ideals (Filters), Prime Fuzzy Ideals (Filters) and Fuzzy Maximal Ideals (Filters) are also provided. Some properties (analogous to the classical theory) are also proved and the notion of Homomorphism from fuzzy lattices as well as the demonstration of some important propositions about it are also provided.

Fuzzy Sets and Systems, 2003

We present a survey on representations of ordered structures by fuzzy sets. Any poset satisfying some ÿniteness condition, semilattice, lattice belonging to a special class, e.g., distributive, Noetherian, complete and others-can be represented by a single function, i.e., by a fuzzy set. Its domain and co-domain are particular subsets of the same structure, and consist of irreducible elements. The representation is minimal in the sense that another representation could not be obtained by replacing the domain of the former by its proper subset. By this approach, the structure itself is uniquely represented by the collection of cuts ordered dually to inclusion.

Fuzzy Sets and Systems, 2007

In this note a new solution of problem of Synthesis of fuzzy sets is presented. In other words, necessary and sufficient conditions are formulated, under which for a given family of subsets F of a set X and a fixed complete lattice L there is a fuzzy set : X → L, such that the collection of cuts of coincides with F. Moreover, it is proved that the general form of lattice-valued fuzzy sets (considering families of cuts) are the type of fuzzy sets having the codomain {0, 1} c for a suitable chosen cardinal c. Problem I. Under which conditions two fuzzy sets : X → L and : X → L have the same families of cuts?

In this paper, we introduce the concept of neutrosophic fuzzy lattice via fuzzy partial ordering. The definition of neutrosophic fuzzy lattice and equipotent are developed with suitable examples. Cartesian product and some equivalent properties of neutrosophic fuzzy lattice are discussed. Also some theorems on neutrosophic fuzzy lattice are developed here.

Computers & Mathematics with Applications, 2011

Fuzzy set closed under fuzzy relation Eigen fuzzy set Fuzzy control Lattice valued fuzzy sets and relations a b s t r a c t Motivated by fuzzy control problems and by some investigations of eigen fuzzy sets, we deal with a closedness of fuzzy sets under fuzzy relations in two ways: in one sense by directly analyzing fuzzy concepts and in the other by investigating the corresponding crisp problems in the cutworthy framework. Our main task is to investigate particular fuzzy functional equations and inequations appearing in this context, which turn out to be essentially connected with fuzzy control problems. We analyze procedures and find solutions of these equations and inequations, pointing to important applications.

Fuzzy relations on the same domain are classified according to the equality of families of cut sets. This equality of fuzzy relations is completely character- ized, not only for unit interval valued fuzzy relations, but more generally, for fuzzy relations whose domain is a (com- plete) lattice. Similar results for fuzzy sets in general are considered in (12). Applications of the results for fuzzy con- gruence relations are also presented.

Fuzzy Sets and Systems, 2003

The aim of the paper is to present a role of fuzzy sets in the theory of ordered structures. Main algebraic properties of cuts of fuzzy sets are given, and a completion of partially ordered sets to complete lattices is described. It turns out that this completion is equivalent with the famous Dedekind-MacNeille completion, but the algorithm presented here is much simpler.

Fuzzy Sets and Systems - FSS, 2010

This paper deals with quantitative domain theory via fuzzy sets. It examines the continuity of fuzzy directed complete posets (dcpos for short) based on complete residuated lattices. First, we show that a fuzzy partial order in the sense of Fan and Zhang and an L-order in the sense of Bělohlávek are equivalent to each other. Then we redefine the concepts of fuzzy directed subsets and (continuous) fuzzy dcpos. We also define and study fuzzy Galois connections on fuzzy posets. We investigate some properties of (continuous) fuzzy dcpos. We show that a fuzzy dcpo is continuous if and only if the fuzzy-double-downward-arrow-operator has a right adjoint. We define fuzzy auxiliary relations on fuzzy posets and approximating fuzzy auxiliary relations on fuzzy dcpos. We show that a fuzzy dcpo is continuous if and only if the fuzzy way-below relation is the smallest approximating fuzzy auxiliary relation.

Fuzzy Sets and Systems, 1983

In this paper a generalisation of fuzzy relations is introduced-fuzzy relations are defined on fuzzy subsets. Properties like ordered reflexivity, symmetry, transitity, transitive closures of such generalised relations and operations on them are discussed.

In this pamplet, some elementary notions of Lattice Theory like poset, least and greatest elements of a poset, (least) upper bound, (greatest) lower bound, maximal and minimal elements, (meet/join) complete (sub) lattice, (weak) sub poset, infinitely distributive lattice, were studied together with examples and some of their important properties which will be very useful, especially for, intuitionistic fuzzy set theory were stated and proved.