NonEmptyPowerSet - Maple Help

Hypergraphs[ExampleHypergraphs]

 NonEmptyPowerSet
 Return the power set of given order

 Calling Sequence NonEmptyPowerSet(S)

Parameters

 S -

Description

 • The command NonEmptyPowerSet(S) returns the hypergraph with the set S as vertex set and  with all non-empty subsets of S as hyperedges.

Examples

 > $\mathrm{with}\left(\mathrm{Hypergraphs}\right):$$\mathrm{with}\left(\mathrm{ExampleHypergraphs}\right):$

Consider the following power set hypergraph.

 > $\mathrm{P4}≔\mathrm{NonEmptyPowerSet}\left(\left\{1,2,3,4\right\}\right)$
 ${\mathrm{P4}}{≔}{\mathrm{< a hypergraph on 4 vertices with 15 hyperedges >}}$ (1)

Draw a graphical representation of this hypergraph.

 > $\mathrm{Draw}\left(\mathrm{P4}\right)$

Compute its minimal hyperedges.

 > $M≔\mathrm{Min}\left(\mathrm{P4}\right);$$\mathrm{Vertices}\left(M\right);$$\mathrm{Hyperedges}\left(M\right)$
 ${M}{≔}{\mathrm{< a hypergraph on 4 vertices with 4 hyperedges >}}$
 $\left[{1}{,}{2}{,}{3}{,}{4}\right]$
 $\left[\left\{{1}\right\}{,}\left\{{2}\right\}{,}\left\{{3}\right\}{,}\left\{{4}\right\}\right]$ (2)

Compute its maximal hyperedges.

 > $M≔\mathrm{Max}\left(\mathrm{P4}\right);$$\mathrm{Vertices}\left(M\right);$$\mathrm{Hyperedges}\left(M\right)$
 ${M}{≔}{\mathrm{< a hypergraph on 4 vertices with 1 hyperedges >}}$
 $\left[{1}{,}{2}{,}{3}{,}{4}\right]$
 $\left[\left\{{1}{,}{2}{,}{3}{,}{4}\right\}\right]$ (3)

Consider this other power set hypergraph.

 > $\mathrm{P8}≔\mathrm{NonEmptyPowerSet}\left(\left\{1,2,3,4,5,6,7,8\right\}\right)$
 ${\mathrm{P8}}{≔}{\mathrm{< a hypergraph on 8 vertices with 255 hyperedges >}}$ (4)

Draw a graphical representation of this hypergraph.

 > $\mathrm{Draw}\left(\mathrm{P8}\right)$

Compute its minimal hyperedges.

 > $M≔\mathrm{Min}\left(\mathrm{P8}\right);$$\mathrm{Vertices}\left(M\right);$$\mathrm{Hyperedges}\left(M\right)$
 ${M}{≔}{\mathrm{< a hypergraph on 8 vertices with 8 hyperedges >}}$
 $\left[{1}{,}{2}{,}{3}{,}{4}{,}{5}{,}{6}{,}{7}{,}{8}\right]$
 $\left[\left\{{1}\right\}{,}\left\{{2}\right\}{,}\left\{{3}\right\}{,}\left\{{4}\right\}{,}\left\{{5}\right\}{,}\left\{{6}\right\}{,}\left\{{7}\right\}{,}\left\{{8}\right\}\right]$ (5)

Compute its maximal hyperedges.

 > $M≔\mathrm{Max}\left(\mathrm{P8}\right);$$\mathrm{Vertices}\left(M\right);$$\mathrm{Hyperedges}\left(M\right)$
 ${M}{≔}{\mathrm{< a hypergraph on 8 vertices with 1 hyperedges >}}$
 $\left[{1}{,}{2}{,}{3}{,}{4}{,}{5}{,}{6}{,}{7}{,}{8}\right]$
 $\left[\left\{{1}{,}{2}{,}{3}{,}{4}{,}{5}{,}{6}{,}{7}{,}{8}\right\}\right]$ (6)

References

 Claude Berge. Hypergraphes. Combinatoires des ensembles finis. 1987,  Paris, Gauthier-Villars, translated to English.
 Claude Berge. Hypergraphs. Combinatorics of Finite Sets.  1989, Amsterdam, North-Holland Mathematical Library, Elsevier, translated from French.
 Charles Leiserson, Liyun Li, Marc Moreno Maza and Yuzhen Xie " Parallel computation of the minimal elements of a poset." Proceedings of the 4th International Workshop on Parallel Symbolic Computation (PASCO) 2010: 53-62, ACM.

Compatibility

 • The Hypergraphs[ExampleHypergraphs][NonEmptyPowerSet] command was introduced in Maple 2024.