Discrete systems such as sets, monoids, groups are familiar categories. The internal strucutre of the latter two is defined by an algebraic operator. In this paper we describe the internal structure of the base set by a closure operator. We illustrate the role of such closure in convex geometries and partially ordered sets and thus suggest the wide applicability of closure systems. Next we deve...

The concepts of fuzzy θ-open (θ-closed)sets and fuzzy θ-closure operator are introduced and discussed in intuitionistic fuzzy topological spaces. As applications of these concepts, certain functions are characterized in terms of intuitionistic fuzzy θ-closure operator.

We construct a Galois connection between closure and interior operators on a given set. All arguments are intuitionistically valid. Our construction is an intuitionistic version of the classical correspondence between closure and interior operators via complement. In classical mathematics, the theory of closure operators and that of interior operators can be derived one from another. In fact, A...

Each relation induces a new closure operator, which is (in the sense of data mining) stronger than or equal to the Galois one. The goal is to give some evidence that the new closure operator is often properly stronger than the Galois one. An easy characterization of the new closure operator as a largest fixed point of an appropriate contraction map leads to a (modest) computer program. Finally,...

A pair (X, ) of a finite set X and a closure operator : 2X → 2X is called a closure space. The class of closure spaces includes matroids as well as antimatroids. Associated with a closure space (X, ), the extreme point operator ex: 2X → 2X is defined as ex(A) = {p|p ∈ A,p / ∈ (A − {p})}. We give characterizations of extreme point operators of closure spaces, matroids and antimatroids, respectiv...

This paper focuses on the basic operations of Chomsky’s languages. The validity and the effectiveness of some closure operations, such as union operator, product operator and Kleene Closure operator, are discussed in detail. The crosstalk problems in Context-Sensitive Languages (CSL) and Phrase Structure Languages (PSL) are analyzed, and a valuable method to solve this problem is presented by s...

Closure is a fundamental property of many discrete systems. Transitive closure in relations has been well studied, e.g. 1,14,6,5], as has geometric closure 8,9] and closure in various kinds of graphs 17,10]. The closed sets of a closure operator illustrate a kind of well-behaved internal structure that is the main theme of this paper. In Section 1, we examine antimatroid closure spaces. In Sect...

The paper presents a new definition of closure operator which encompasses the standard Dikranjan-Giuli notion, as well as the Bourn-Gran notion of normal closure operator. As is well known, any two closure operators C,D in a category may be composed in two ways: For a subobject M → X one may consider DX(CXM) or DCX(M)(M) as the value at M of a new closure operator D ·C or D ∗C, respectively. Th...