## 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.