Topological and Algebraic Structures of Atanassov's Intuitionistic Fuzzy-Values Space
This work addresses foundational issues in fuzzy set theory for researchers in mathematics and computer science, but it is incremental as it builds on existing frameworks to answer specific open problems.
The paper tackles the problem of characterizing the algebraic and topological structures of intuitionistic fuzzy values (IFVs) by proving that two linear orders yield the same algebraic structure, introducing a new operator that acts as a strong negation, and showing that the space is a complete lattice and Kleene algebra. It also demonstrates that the topological space is compact and connected but not separable or metrizable, partially answering open problems from Atanassov's work, and constructs an isomorphism between IFVs and q-rung orthopedic fuzzy values (q-ROFVs), with an application to a pattern recognition problem for classifying building materials.
We prove that the space of intuitionistic fuzzy values (IFVs) with a linear order based on a score function and an accuracy function has the same algebraic structure as the one induced by a linear order based on a similarity function and an accuracy function. By introducing a new operator for IFVs via the linear order based on a score function and an accuracy function, we show that such an operator is a strong negation on IFVs. Moreover, we observe that the space of IFVs is a complete lattice and a Kleene algebra with the new operator. We also demonstrate that the topological space of IFVs with the order topology induced by the above two linear orders is not separable and metrizable but compact and connected. From some new perspectives,our results partially answer three open problems posed by Atanassov [Intuitionistic Fuzzy Sets: Theory and Applications, Springer, 1999] and [On Intuitionistic Fuzzy Sets Theory, Springer, 2012]. Furthermore, we construct an isomorphism between the spaces of IFVs and q-rung orthopedic fuzzy values (q-ROFVs) under the corresponding linear orders. To this end, we introduce the concept of admissible similarity measures with particular orders for IFSs, extending the existing definition of the similarity measure for IFSs, and construct an admissible similarity measure with a linear order based on a score function and an accuracy function, which is effectively applied to a pattern recognition problem about the classification of building materials.