A T-fuzzy equivalence relation is a fuzzy binary relation on a set X which is reflexive, symmetric and T-transitive for a t-norm T. Recently, Mesiar et al. [R. Mesiar, B. Reusch, H. Thiele, Fuzzy equivalence relations and fuzzy partitions, J. Multi-Valued Logic Soft Comput. 12 (2006) 167–181] have generalised the t-norm T to any general conjunctor C and investigated the minimal assumptions requ...

Kernel methods and rough sets are two general pursuits in the domain of machine learning and intelligent systems. Kernel methods map data into a higher dimensional feature space, where the resulting structure of the classification task is linearly separable; while rough sets granulate the universe with the use of relations and employ the induced knowledge granules to approximate arbitrary conce...

A method to compute the transitive closure, a transitive opening and a transitive approximation of a reflexive and symmetric fuzzy relation is given. Other previous methods in literature compute just the transitive closure, some transitive approximations or some transitive openings. The proposed algorithm computes the three different similarities that approximate a proximity for the computation...

Some existence theorems for the mixed variational-like inequality for fuzzy mappings FMVLIP in a reflexive Banach space are established. Further, the auxiliary principle technique is used to suggest a novel and innovative iterative algorithm for computing the approximate solution. Consequently, not only the existence of solutions of the FMVLIP is shown, but also the convergence of iterative seq...

The paper continues a study of extensional formulas and their representation by normal forms. The notion of extensional formula is extended to the case when extensionality is considered with respect to a binary reflexive relation. It has been proved that extensional formulas can be represented by normal forms of infinite type and “approximated” by normal forms of finite type. All proofs are giv...

Let F be the set of all fuzzy numbers. In this study, we embed F into a nonseparable and non-reflexive unital Banach algebra A containing F by means of a norm defined on A, which enable us to consider F as a subset of A. Besides, the Banach lattice structure of the space A according to a partial order, we define on A has been touched on.

Let R be a commutative Noetherian ring. The k-torsionless modules are defined in [7] as a generalization of torsionless and reflexive modules, i.e., torsionless modules are 1-torsionless and reflexive modules are 2-torsionless. Some properties of torsionless, reflexive, and k-torsionless modules are investigated in this paper. It is proved that if M is an R-module such that G-dimR(M)

− Spatial data mining knows a more and more important interest. Fundamental processes of spatial data mining are in particular clustering and structural patterns detection. These processes are influenced strongly by the concept of proximity or neighborhood. This paper introduces some structures to the construction of a spatial data mining integrating fuzzy structural primitives and propose to o...

The problem of embedding incomplete into complete relations has been an important topic of research in the context of crisp relations and their applications. Several variations of the acclaimed Szpilrajn’s theorem have been provided, inclusive of the case when some order conditions between elements are imposed on the extension. We extend the analysis of that topic by Alcantud to the fuzzy case....

Two ways to approximate a proximity relation R (i.e. a reflexive and symmetric fuzzy relation) by a T -transitive one where T is a continuous archimedean t-norm are given. The first one aggregates the transitive closure R of R with a (maximal) T -transitive relation B contained in R. The second one modifies the values of R or B to better fit them with the ones of R.