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,...

We define and study the spaces Mp(R), 1 ⩽ p ⩽ +∞, that are of DLp -type. Using the harmonic analysis related to the Fourier transform connected with the Riemann-Liouville operator, we give a new characterization of the dual space M ′ p(R) and we describe its bounded subsets. Next, we define a convolution product in M ′ p(R)×Mr(R), 1 ⩽ r ⩽ p < +∞, whereMr(R) is the closure of the space S∗(R) in ...

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...

The notion of a transitive closure of a fuzzy relation is very useful for clustering in pattern recognition, for fuzzy databases, etc. It is based on translating the standard definition of transitivity and transitive closure into fuzzy terms. This definition works fine, but to some extent it does not fully capture our understanding of transitivity. The reason is that this definition is based on...

Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This paper has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library 1 We show how some c...

The es-splitting operation on binary matroids is a natural generalization of Slater's &lt;em&gt;n&lt;/em&gt;-line splitting graphs. In this paper, we characterize the closure operator matroid M&lt;sup&gt;e&lt;/sup&gt;&lt;sub&gt;X&lt;/sub&gt; in terms original M. We also describe ats and hyperplanes bi- nary hyperplanes, respectively

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...

For any ordered set P, the join dense completions of P form a complete lattice K(P) with least element O(P), the lattice of order ideals of P, and greatest element M(P), the Dedekind-MacNeille completion of P. The latticeK(P) is isomorphic to an ideal of the lattice of all closure operators on the lattice O(P). Thus it inherits some local structural properties which hold in the lattice of closu...