Some Property of an Ultrafilter and Graph parameters on Connectivity System

An ultrafilter is a maximal filter on a set, playing a crucial role in set theory and topology for rigorously handling limits, convergence, and compactness. A connectivity system is defined as a pair (X, f), where X is a finite set and f is a symmetric submodular function. Understanding the duality in these parameters helps to elucidate the relationship between different decompositions and measures of a graph's complexity. In this paper, we delve into ultrafilters on connectivity systems, applying Tukey's Lemma to these systems. Additionally, we explore prefilters, ultra-prefilters, and subbases within the context of connectivity systems. Furthermore, we introduce and investigate new parameters related to width, length, and depth.