Neighborhood system in concept lattice

Abstract

In this thesis, we apply and generalize the notion of neighborhood system from topology to study the relation between the concepts in a concept lattice. We classify all concepts in the concept lattice into various classes by seeking similar characters or properties of their attributes. Any element in the concept lattice is associated with a family of subsets o f the concept lattice. This family is called a neighborhood system of the element. Each subset in the neighborhood system is called a neighborhood of the element. A concept in some neighborhood of the fixed element in the concept lattice is interpreted to be somewhat near or adjacent to the element Two concepts in a same neighborhood are considered to be somewhat indiscernible or at least not noticeably distinguishable. We introduce three different neighborhood systems NSi, NSa and NS3 . For the first type NSi, a concept is said to be in a neighborhood of another concept in the concept lattice if it is a subconcept or a superconcept of the other. For die second type NS2 , a concept is said to be in a neighborhood of another concept if the two concepts have some common attributes. For the third type NS3 , a concept is said to be in the neighborhood o f another concept if every object in the concept shares some attribute with some object in the other concept. We prove that (see document) Examples are given and properties o f the neighborhood systems are discussed.