1.1 What is a Group? A group is a set of objects with an associated operation which satisfies certain properties. For a group G which is a set of elements {T, T ′, T ′′, ...} with operator ◦, we require: 1. Closure. ∀T, T ′ ∈ G : T ◦ T ′ ∈ G. This property of closure should be familiar to anyone with a linear algebra background, as a vector space also fulfills this property, with addition as th...

We extend von Neumann’s Double Commutant Theorem to the setting of nonselfadjoint operator algebras A, while restricting the notion of commutants of a subset S of A to those operators in A which commute with every operator in S. If A is a completely distributive commutative subspace lattice algebra acting on a Hilbert space H, we obtain an alternate characterization (to those of Erdos–Power and...

The generalised structural arrangements of various bimolecule such as carbohydrates, proteins etc are based on the rules ofČech closure operator. In this paper we discuss the relationship betweeň Cech closure space and structural configuration of proteins.

We argue that closure operators are fundamental tools for the study of relationships between order structures and their sequence representations. We also propose and analyse a closure operator for interval order structures.

This paper is a study about the relationships among topologies and intuitionistic fuzzy topology induced, respectively, by approximation operators and an intuitionistic fuzzy approximation operator associated with an approximation space (X,R), when the relation R on X is precisely reflexive and transitive. In particular, we consider an intuitionistic fuzzy approximation operator on an approxima...

This remarkable little theorem and related phenomena have been the concern of many authors. Apart from the mysterious appearance of the number 14, the attraction of this theorem is that it is simple to state and can be examined and proved using concepts available after any first encounter with topology. The goal of this article is both to provide an original investigation into variations of the...

A new class of harmonic multivalent functions defined by an integral operator is introduced. Coefficient inequalities, extreme points, distortion bounds, inclusion results and closure under an integral operator for this class are obtained .

Let G be the Klein four-group and let k an arbitrary field of characteristic 2. A classification indecomposable kG-modules is known. We calculate relative cohomology groups Hχi(G,N) for every kG-module N, where χ set proper subgroups in G. This extends work Pamuk Yalcin to with non-trivial coefficients. also show that all cup products strictly positive degree Hχ*(G,k) are trivial.

In a former paper, the algorithm NextClosures for computing the set of all formal concepts as well as the canonical base for a given formal context has been introduced. Here, this algorithm shall be generalized to a setting where the data-set is described by means of a closure operator in a complete lattice, and furthermore it shall be extended with the possibility to handle constraints that ar...

We define the notion of rational closure in the context of Description Logics extended with a tipicality operator. We start from ALC +T, an extension of ALC with a typicality operator T : intuitively allowing to express concepts of the form T(C), meant to select the “most normal” instances of a concept C. The semantics we consider is based on rational model. But we further restrict the semantic...