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The investigation of width parameters in both graph and algebraic contextshas attracted considerable interest. Among these parameters, the linear branchwidth has emerged as a crucial measure. In this concise paper, we explore theconcept of linear decomposition, specifically focusing on the single filter ina connectivity system. Additionally, we examine the relevance of matroids,antimatroids, and greedoids in the context of connectivity systems. Our primaryobjective in this study is to shed light on the impediments to lineardecomposition from multiple perspectives.

Matroid, Ideal, Ultrafilter, Tangle, and so on: Reconsideration of Obstruction to linear decomposition