A Binary Intuitionistic Fuzzy Relation: Some New Results, a General Factorization, and Two Properties of Strict Components

Abstract

We establish, by means of a large class of continuous t-representable intuitionistic fuzzy t-conorms, a factorization of an intuitionistic fuzzy relation IFR into a unique indifference component and a family of regular strict components. This result generalizes a previous factorization obtained by Dimitrov 2002 with the max, min intuitionistic fuzzy t-conorm. We provide, for a continuous t-representable intuitionistic fuzzy t-norm T, a characterization of the T-transitivity of an IFR. This enables us to determine necessary and sufficient conditions on a T-transitive IFR R under which a strict component of R satisfies pos-transitivity and negative transitivity.

Difference and symmetric difference for intuitionistic fuzzy sets

“…Proposition 10 (see [13]). Let S and be such that, for all a, b ∈ The following examples of fuzzy t-norms and fuzzy t-conorms belonging to a family called Frank t-norms and Frank t-conorms will commonly be referred to in this study (see [10,13]):…”

“…where S M , S P , S L are the maximum, probabilistic sum and Lukasiewicz fuzzy t-conorms, respectively (see [9,10,13,16,17]). .…”

“…Here, we introduce the basic concepts of intuitionistic fuzzy sets, recall the definitions and examples of some intuitionistic fuzzy operators and operations ( [5,13,17]).…”

“…Definition 4 (t-Representable intuitionistic fuzzy t-norm and t-conorm (see [5,6,13,17])). An intuitionistic fuzzy t-norm T (respectively intuitionistic fuzzy t-conorm J ) is t-representable if there exists a fuzzy t-norm and a fuzzy t-conorm S (respectively a fuzzy t-conorm S and a fuzzy t-norm ) such that, for all…”

“…[10,13,17]: for all a, b ∈ [0, 1]:1. Fuzzy R-implication and fuzzy co-implication associated with M and S M are respectively given by I M (a, b) Fuzzy R-implication and fuzzy co-implication associated with P and S P are respectively given byI P (a, b) =    if a ≤ b, b a ,if a > b.…”