There are many symmetries in intuitionistic fuzzifying topology. In the present paper, the notion of intuitionistic fuzzifying topology as an extension of fuzzifying topology and a preliminary of the research on bi-intuitionistic fuzzy topology is introduced. A theory of intuitionistic fuzzy topology with the semantic method of intuitionistic fuzzy logic is established.

In this article, we introduce two new rough set models based on the concept of double fuzzy relations. These models are called optimistic and pessimistic multi-granulation double fuzzy rough sets. We discuss their properties and explore the relationship between these new models and double fuzzy rough sets. Our study focuses on the lower and upper approximations of these models, which generalize the conventional rough set model. In addition, we suggest that the development of the multi-granulation double fuzzy rough set model is significant for the generalization of the rough set model.

Many existing control techniques proposed in the literature tend to overlook faults and physical limitations in the systems, which significantly restricts their applicability to practical, real-world systems. Consequently, there is an urgent necessity to advance the control and synchronization of such systems in real-world scenarios, specifically when faced with the challenges posed by faults and physical limitations in their control actuators. Motivated by this, our study unveils an innovative control approach that combines a neural network-based sliding mode algorithm with fuzzy logic systems to handle nonlinear systems. This proposed controller is further enhanced with an intelligent observer that takes into account potential faults and limitations in the control actuator, and it integrates a fuzzy logic engine to regulate its operations, thus reducing system chatter and increasing its adaptability. This strategy enables the system to maintain regulation in the face of control input constraints and faults and ensures that the closed-loop system will achieve convergence within a finite-time frame. The detailed explanation of the control design confirms its finite-time stability. The robust performance of the proposed controller applied to autonomous and non-autonomous systems grappling with control input limitations and faults demonstrates its effectiveness.

This paper aims to define fuzzy pretopogenous structure based on way below relation (or an L-fuzzifying pretopogenous structure (LFPT structure, for short)) and study some of its properties. Also, the concepts of L-fuzzifying pre-neighborhood, L-fuzzifying pre-interior, and L-fuzzifying pre-closure operators are established and we used these concepts to build an L-fuzzifying topology. Furthermore, a natural link is established between L-fuzzifying pretopogenous and L-fuzzifying topology. Finally, the maps between L-fuzzifying pretopogenous structures and initial fuzzifying structures are investigated.

To define the optimistic multiple-granulation (OMG) supra interior operator and the OMG-supra closure operator associated with a set A in a multiple-granulation approximation space, we used the concepts of the OMG-lower set OL (A) and the OMG-upper set OU (A). These operators create an approximate topological space known as optimistic multiple-granulation supratopological spaces. OMG-supra-open and OMG-supra-closed approximation sets are also defined, followed by OMG-supra-continuous approximation and OMG-irresolute approximation.