in this paper we will generalize  some of known results on the tight closure of an ideal to the tight closure of an ideal relative to a module .

This paper is devoted to the study of nondeterministic closure automata, that is, nondeterministic finite automata (nfas) equipped with a strict closure operator on the set of states and continuous transition structure. We prove that for each regular language L there is a unique minimal nondeterministic closure automaton whose underlying nfa accepts L. Here minimality means no proper sub or quo...

Covering-based rough set theory is a useful tool to deal with inexact, uncertain or vague knowledge in information systems. Topology, one of the most important subjects in mathematics, provides mathematical tools and interesting topics in studying information systems and rough sets. In this paper, we present the topological characterizations to three types of covering approximation operators. F...

‎In this paper we try to extend geometric concepts in the context of operator valued tensors‎. ‎To this end‎, ‎we aim to replace the field of scalars $ mathbb{R} $ by self-adjoint elements of a commutative $ C^star $-algebra‎, ‎and reach an appropriate generalization of geometrical concepts on manifolds‎. ‎First‎, ‎we put forward the concept of operator-valued tensors and extend semi-Riemannian...

Double crises of chaotic oscillators in the presence of fuzzy uncertainty are studied by means of the fuzzy generalized cellmappingmethod.A fuzzy chaotic attractor is characterized by its global topology and membership distribution. A fuzzy crisis implies a simultaneous sudden change both in the topology of the chaotic attractor and in its membership distribution. It happenswhen a fuzzy chaotic...

Li and Pedrycz [Y. M. Li, W. Pedrycz, Fuzzy finite automata and fuzzy regular expressions with membership values in lattice ordered monoids, Fuzzy Sets and Systems 156 (2005) 68–92] have proved fundamental results that provide different equivalentways to represent fuzzy languageswithmembership values in a lattice-orderedmonoid, and generalize the well-known results of the classical theory of fo...

In this article we formalize the Kuratowski closure-complement result: there is at most 14 distinct sets that one can produce from a given subset A of a topological space T by applying closure and complement operators and that all 14 can be obtained from a suitable subset of R, namely KuratExSet = {1} ∪Q(2, 3) ∪ (3, 4) ∪ (4,∞). The second part of the article deals with the maximal number of dis...

In an 〈E,M 〉-category X for sinks, we identify necessary conditions for Galois connections from the power collection of the class of (composable pairs) of morphisms in M to factor through the “lattice” of all closure operators on M , and to factor through certain sublattices. This leads to the notion of regular closure operator. As one byproduct of these results we not only arrive (in a novel w...

We study interior operators and interior structures in a fuzzy setting. We investigate systems of “almost open” fuzzy sets and the relationships to fuzzy interior operators and fuzzy interior systems.