A study of lattice of c-structures and homogeneous C-spaces

Abstract

The concept of connected sets is defined in several mathematical fields, including topology, graph theory, and their fuzzy analogues. Connectivity plays a crucial role in digital image processing. However, the approaches to connectedness in topology and graph theory do not necessarily coincide, but the compatibility of these two connectedness is essential as discrete image can be obtained from continuous scenes. To overcome these limitations, an integrated approach is required. All meaningful notions of connectivity share the following properties: (i) The empty set and the points are connected. (ii) The union of overlapping connected objects is connected. In 1983, R. Borger introduced an axiomatic approach to connectivity: the theory of connectivity classes, or c-structures which adopts properties (i) and (ii) as axioms. This approach broadens the notion of connectedness in both graph theory and topology. Pioneering contributions in this area have been made by J. Serra, S. Dugowson, J. Muscat and D. Buhagiar, H. J. A. M. Heijmans and C. Ronse, among others. ) The theoretical development of the theory of connectivity classes is essential for its advancement in application-based studies. The primary goal of this thesis is to present novel contributions to the theory of c-spaces. The thesis mainly examines the properties of the lattice of c-structures and homogeneous c-spaces. The concept of simple expansion is defined and also study the relationship between simple expansion and upper neighbors of a c-structure. The automorphism group of the lattice of e-structures and the fixed points of the automorphism group of the lattice of c-structures are determined. We introduce and examine three distinct types of homogeneity in c-spaces namely; n-homogeneity, strongly n-homogeneity, and local homogeneity. Several characteristics of hereditarily homogeneous c-spaces are examined. Additionally, hereditarily homogeneous c-spaces are characterized in terms of c-automorphisms. Extending the concept of c-spaces, we define fuzzy c-spaces. Subsequently, it is shown that the collection of all fuzzy cstructures on a given set forms a complete lattice, and its lattice properties are explored.