Dependency relations

Abstract

In this paper, we introduce a notion of dependency between subsets of an arbitrary fixed non-empty set \(\Omega \). To be more detailed, we introduce a preorder \(\leftarrow \) on the power set \(\mathcal {P}(\Omega )\) having the further property that \(B \leftarrow A\) if and only if \(\{b\} \leftarrow A\) for any \(b \in B\). We shall argue that this relation generalizes well-studied notions of dependence occurring in such fields as linear algebra, topology, and combinatorics. Furthermore, we show that this relation is characterized by two set operators whose fixed points have interesting geometric and order-theoretic properties. After giving some some elementary results about such a dependency relation, we provide some specific examples taken from graph theory. An interesting property we will provide consists of the possibility to characterize partial orders on a finite lattice in terms of a suitable dependency relation. Finally, we introduce and analyze some specific classes of dependency relations, namely attractive and anti-attractive dependency relations.

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Acknowledgements

We are extremely thankful to the unknown reviewers whose thorough objections and suggestions have been very useful in order to improve the quality of our paper.

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Correspondence to G. Chiaselotti.

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Chiaselotti, G., Infusino, F. & Oliverio, P.A. Dependency relations. Boll Unione Mat Ital 12, 525–548 (2019). https://doi.org/10.1007/s40574-018-00188-z

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Keywords

  • Abstract dependency
  • Closure systems
  • Abstract simplicial complexes
  • Graphs

Mathematics Subject Classification

  • Primary 08A02
  • 08A05
  • 06A06
  • Secondary 05C50
  • 05C75