Categories of interval-valued fuzzy sets

Tatra Mountains Mathematical Publications, 2016

In this paper, a new relation for the set of interval-valued fuzzy relations is introduced. This relation is an interval order for the family of intervals and for the family of interval-valued fuzzy relations in a given set, it has the reflexivity property. Consequences of considering such a relation are studied in the context of operations on interval-valued fuzzy relations. A new transitivity property, namely possible T-transitivity is studied (pos-T-transitivity for short). This transitivity property is connected with the new relation proposed in this paper. Preservation of this type of transitivity by some operations is also discussed.

Handbook of Granular Computing, 2008

2019

This work aims to introduce other approaches to the interval-valued fuzzy logic. These new approaches were inspired by Lodwick and Chalco's works on constraint intervals. These constraint intervals were used in this thesis to extend the fuzzy operators into two modes, named Single-Level Constrained Interval Operators and Constrained Interval Operators and studied their properties. A new algebra, called SBCI algebra, which arises from the intervalization of BCI-algebras, is also introduced. These algebras aims to be the algebraic model for intervalvalued fuzzy logics, which take into account the notion of correctness. A new class of fuzzy implications, called (T, N)-implications has also been studied. The author investigated the behavior of the BCI/SBCI algebras and (T, N)-implications.

The aim of this paper is 1. to introduce the notions of, an interval valued f-set with truth values in a complete lattice of closed intervals or a simply a cloci over an arbitrary a complete lattice , L called an L -interval valued f-set or simply an L -ivf-set, an L -interval valued fsubset and to introduce an interval valued f-map between an L -interval valued f-set and an M -interval valued f-set where the complete lattice L may possibly be different from the complete lattice , M an M -interval valued f-image of an L -interval valued f-subset under an interval valued f-map and an L -interval valued f-inverse image of an M -interval valued f-subset under an interval valued f-map, and 2. to study the standard (lattice) algebraic properties of, all L -interval valued f-subsets of an L -interval valued f-set, all M -interval valued f-images of Linterval valued f-subsets under an interval valued f-map and of all L -interval valued f-inverse images of M -interval valued f-subsets under an interval valued f-map, generalizing the Theory of f-Sets.

The aim of this paper is to introduce the notion of interval-valued fuzzy subspace with its flags and interval-valued fuzzy n-normed linear space. We define the operations intersection, sum, directsum and tensor product of intervalvalued fuzzy subspaces and obtain their corresponding flags. Further we provide some results on interval-valued fuzzy n-normed linear space.

Joint Proceedings of the 19th World Congress of the International Fuzzy Systems Association (IFSA), the 12th Conference of the European Society for Fuzzy Logic and Technology (EUSFLAT), and the 11th International Summer School on Aggregation Operators (AGOP), 2021

In this work, we provide a study on the representability of interval-valued fuzzy connectives considering partial and admissible orders. Our approach considers those orders based on injective aggregation functions which allowed the construction of intervalvalued operators. An immediate result is the construction of the implication used in the Quantum Logic, known as QL-implication, that in our proposal is generalised for intervals, given by means of interval-valued t-(co)norms and interval-valued fuzzy negations.

Fuzzy Sets and Systems, 2006

Among the various extensions to the common [0, 1]-valued truth degrees of "traditional" fuzzy set theory, closed intervals of [0, 1] stand out as a particularly appealing and promising choice for representing imperfect information, nicely accommodating and combining the facets of vagueness and uncertainty without paying too much in terms of computational complexity. From a logical point of view, due to the failure of the omnipresent prelinearity condition, the underlying algebraic structure L I falls outside the mainstream of the research on formal fuzzy logics (including MV-, BL-and MTL-algebras), and consequently so far has received only marginal attention. This comparative lack of interest for interval-valued fuzzy logic has been further strengthened, perhaps, by taking for granted that its algebraic operations amount to a twofold application of corresponding operations on the unit interval. Abandoning that simplifying assumption, however, we may find that L I reveals itself as a very rich and noteworthy structure allowing the construction of complex and surprisingly well-behaved logical systems. Reviewing the main advances on the algebraic characterization of logical operations on L I , and relating these results to the familiar completeness questions (which remain as major challenges) for the associated formal fuzzy logics, this paper paves the way for a systematic study of interval-valued fuzzy logic in the narrow sense.

International Journal of Approximate Reasoning, 2004

With the demand for knowledge-handling systems capable of dealing with and distinguishing between various facets of imprecision ever increasing, a clear and formal characterization of the mathematical models implementing such services is quintessential. In this paper, this task is undertaken simultaneously for the definition of implication within two settings: first, within intuitionistic fuzzy set theory and secondly, within interval-valued fuzzy set theory. By tracing these models back to the underlying lattice that they are defined on, on one hand we keep up with an important tradition of using algebraic structures for developing logical calculi (e.g. residuated lattices and MV algebras), and on the other hand we are able to expose in a clear manner the two modelsÕ formal equivalence. This equivalence, all too often neglected in literature, we exploit to construct operators extending the notions of classical and fuzzy implication on these structures; to initiate a meaningful classification framework for the resulting operators, based on logical and extra-logical criteria imposed on them; and finally, to re(de)fine the intuititive ideas giving rise to both approaches as models of imprecision and apply them in a practical context.

atlantis-press.com

In this paper, we introduced intervalvalued fuzzy matrices (IVFMs) as the generalization of interval-valued fuzzy sets. Some essential unary and binary operations of IVFM and some special types of IVFMs ie, symmetric, reflexive, transitive and idempotent, ...

Mathematics, 2021

Multiple definitions have been put forward in the literature to measure the differences between two interval-valued fuzzy sets. However, in most cases, the outcome is just a real value, although an interval could be more appropriate in this environment. This is the starting point of this contribution. Thus, we revisit the axioms that a measure of the difference between two interval-valued fuzzy sets should satisfy, paying special attention to the condition of monotonicity in the sense that the closer the intervals are, the smaller the measure of difference between them is. Its formalisation leads to very different concepts: distances, divergences and dissimilarities. We have proven that distances and divergences lead to contradictory properties for this kind of sets. Therefore, we conclude that dissimilarities are the only appropriate measures to measure the difference between two interval-valued fuzzy sets when the outcome is an interval.