F-transform, fuzzy rough sets, fuzzy topology, fuzzy approximation space

In mathematics, the importance of various kinds of transforms such as Fourier, Laplace, integral, wavelet is well-known and well-documented. Recently, a number of researchers have initiated the study of (fuzzy) F-transforms based on an arbitrary L-fuzzy partition of a universe, where L is a complete residuated lattice. F-transforms encompass the above mentioned transforms as well as approximation methods based on the fuzzy IF-THEN rules studied in fuzzy modeling. From an application point of view, this theory has turned out to be useful in denoising, scheduling, time series, coding/decoding of images, numerical solutions of partial differential equations and solution of wave equation.Among these studies, a categorical study of L-partitions of an arbitrary universe is presented by J. Močkoř.In this talk, we will be focused on various spaces where a fuzzy partition is connected with particular structures as fuzzy approximation spaces, fuzzy rough sets, fuzzy topologies, etc. All these structures are characterized using various languages, so that is not easy to understand their interrelationship. We will explain a common characterization of say, fuzzy structured spaces, using the so called operator point of view. We stem from the idea that all these spaces are based on the abstract notion of ``closeness'', and that they are introduced using axiomatically expressed properties of this notion.