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5.5: Partial Order

  • Page ID
    181197
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    A relation \(R\) on a set is a Partial Order (PO) \(\prec\) if it satisfies the reflexive, antisymmetric, and transitive properties. A poset is a set with a partial order relation. For example, the following set of numbers with a relation given by divisibility is a poset.

    A = Set([1, 2, 3, 4, 5, 6, 8])
    
    R = [(a, b) for a in A for b in A if a.divides(b)]
    
    D = DiGraph(R, loops=True)
    
    plot(D)

    A Hasse diagram is a simplified visual representation of a poset. Unlike a digraph, the relative position of vertices has meaning: if \(x\) relates to \(y\), then the vertex \(x\) appears lower in the drawing than the vertex \(y\). Self-loops are assumed and not shown. Similarly, the diagram assumes the transitive property and does not explicitly display the edges that are implied by the transitive property.

    Note

    Partial orders and Hasse diagrams help analyze task dependencies in scheduling applications.


    5.5: Partial Order is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by LibreTexts.

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