GENEOs, compactifications, and graphs

Our objective in this thesis is to study the pseudo-metric and topological structure of the space of group equivariant non-expansive operators (GENEOs). We introduce the notions of compactification of a perception pair, collectionwise surjectivity, and compactification of a space of GENEOs. We obtain some compactification results for perception pairs and the space of GENEOs. We show that when the data spaces are totally bounded and endow the common domains with metric structures, the perception pairs and every collectionwise surjective space of GENEOs can be embedded isometrically into the compact ones through compatible embeddings. An important part of the study of topology of the space of GENEOs is to populate it in a rich manner. We introduce the notion of a generalized permutant and show that this concept too, like that of a permutant, is useful in defining new GENEOs. We define the analogues of some of the aforementioned concepts in a graph theoretic setting, enabling us to use the power of the theory of GENEOs for the study of graphs in an efficient way. We define the notions of a graph perception pair, graph permutant, and a graph GENEO. We develop two models for the theory of graph GENEOs. The first model addresses the case of graphs having weights assigned to their vertices, while the second one addresses weighted on the edges. We prove some new results in the proposed theory of graph GENEOs and exhibit the power of our models by describing their applications to the structural study of simple graphs. We introduce the concept of a graph permutant and show that this concept can be used to define new graph GENEOs between distinct graph perception pairs, thereby enabling us to populate the space of graph GENEOs in a rich manner and shed more light on its structure.