  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.
