The command Max(H) returns the hypergraph L whose vertex set is the vertex set of H and whose hyperedges are the hyperedges of H which are maximal w.r.t. inclusion.

Therefore, the hypergraph L  is the partial hypergraph of H whose hyperedges are not comparable w.r.t. inclusion and such that every hyperedge of H is contained in one hyperedge of L.

Hence, the hyperedges of L are the maximla elements of the partially ordered set (poset) whose vertices are the hyperedges of H and whose ordering is the set-theoretic inclusion.

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

$\left[\mathrm{Fan}\,\mathrm{Kuratowski}\,\mathrm{Lovasz}\,\mathrm{NonEmptyPowerSet}\,\mathrm{RandomHypergraph}\right]$

$H≔\mathrm{Hypergraph}\left(\left[1\,2\,3\,4\,5\,6\,7\right]\,\left[\left\{1\,2\,3\right\}\,\left\{2\,3\right\}\,\left\{4\right\}\,\left\{3\,5\,6\right\}\right]\right)$

$H≔\mathrm{<\; a\; hypergraph\; on\; 7\; vertices\; with\; 4\; hyperedges\; >}$

$\mathrm{Vertices}\left(H\right)\&semi;$$\mathrm{Hyperedges}\left(H\right)$

$\left[1\&comma;2\&comma;3\&comma;4\&comma;5\&comma;6\&comma;7\right]$

$\left[\left\{4\right\}\&comma;\left\{2\&comma;3\right\}\&comma;\left\{1\&comma;2\&comma;3\right\}\&comma;\left\{3\&comma;5\&comma;6\right\}\right]$

$\mathrm{Draw}\left(H\right)$

$M≔\mathrm{Max}\left(H\right)\&semi;$$\mathrm{Vertices}\left(M\right)\&semi;$$\mathrm{Hyperedges}\left(M\right)\&semi;$$\mathrm{Draw}\left(M\right)$

$M≔\mathrm{<\; a\; hypergraph\; on\; 7\; vertices\; with\; 3\; hyperedges\; >}$

$\left[1\&comma;2\&comma;3\&comma;4\&comma;5\&comma;6\&comma;7\right]$

$\left[\left\{4\right\}\&comma;\left\{1\&comma;2\&comma;3\right\}\&comma;\left\{3\&comma;5\&comma;6\right\}\right]$

$M≔\mathrm{Min}\left(H\right)\&semi;$$\mathrm{Vertices}\left(M\right)\&semi;$$\mathrm{Hyperedges}\left(M\right)\&semi;$$\mathrm{Draw}\left(M\right)$

$M≔\mathrm{<\; a\; hypergraph\; on\; 7\; vertices\; with\; 3\; hyperedges\; >}$

$\left[1\&comma;2\&comma;3\&comma;4\&comma;5\&comma;6\&comma;7\right]$

$\left[\left\{4\right\}\&comma;\left\{2\&comma;3\right\}\&comma;\left\{3\&comma;5\&comma;6\right\}\right]$

$T≔\mathrm{Transversal}\left(H\right)\&semi;$$\mathrm{Vertices}\left(T\right)\&semi;$$\mathrm{Hyperedges}\left(T\right)\&semi;$$\mathrm{Draw}\left(T\right)$

$T≔\mathrm{<\; a\; hypergraph\; on\; 7\; vertices\; with\; 3\; hyperedges\; >}$

$\left[1\&comma;2\&comma;3\&comma;4\&comma;5\&comma;6\&comma;7\right]$

$\left[\left\{3\&comma;4\right\}\&comma;\left\{2\&comma;4\&comma;5\right\}\&comma;\left\{2\&comma;4\&comma;6\right\}\right]$

$\mathrm{TT}≔\mathrm{Transversal}\left(T\right)\&semi;$$\mathrm{Hyperedges}\left(\mathrm{TT}\right)\&semi;$$\mathrm{AreEqual}\left(\mathrm{TT}\&comma;\mathrm{Min}\left(H\right)\right)$

$\mathrm{TT}≔\mathrm{<\; a\; hypergraph\; on\; 7\; vertices\; with\; 3\; hyperedges\; >}$

$\left[\left\{4\right\}\&comma;\left\{2\&comma;3\right\}\&comma;\left\{3\&comma;5\&comma;6\right\}\right]$

$\mathrm{true}$

$\mathrm{K3}≔\mathrm{Kuratowski}\left(\left\{1\&comma;2\&comma;3\&comma;4\&comma;5\right\}\&comma;3\right)$

$\mathrm{K3}≔\mathrm{<\; a\; hypergraph\; on\; 5\; vertices\; with\; 10\; hyperedges\; >}$

$\mathrm{AreEqual}\left(\mathrm{Transversal}\left(\mathrm{K3}\right)\&comma;\mathrm{K3}\right)$

$\mathrm{true}$

$R≔\mathrm{RandomHypergraph}\left(5\&comma;15\right)\&semi;$$\mathrm{Hyperedges}\left(R\right)$

$R≔\mathrm{<\; a\; hypergraph\; on\; 5\; vertices\; with\; 15\; hyperedges\; >}$

$\left[\left\{5\right\}\&comma;\left\{1\&comma;2\right\}\&comma;\left\{2\&comma;3\right\}\&comma;\left\{1\&comma;4\right\}\&comma;\left\{2\&comma;4\right\}\&comma;\left\{1\&comma;5\right\}\&comma;\left\{2\&comma;5\right\}\&comma;\left\{3\&comma;5\right\}\&comma;\left\{1\&comma;3\&comma;4\right\}\&comma;\left\{1\&comma;3\&comma;5\right\}\&comma;\left\{2\&comma;3\&comma;5\right\}\&comma;\left\{1\&comma;4\&comma;5\right\}\&comma;\left\{3\&comma;4\&comma;5\right\}\&comma;\left\{1\&comma;2\&comma;3\&comma;5\right\}\&comma;\left\{1\&comma;2\&comma;4\&comma;5\right\}\right]$

$M≔\mathrm{Min}\left(R\right)\&semi;$$\mathrm{Hyperedges}\left(M\right)$

$M≔\mathrm{<\; a\; hypergraph\; on\; 5\; vertices\; with\; 5\; hyperedges\; >}$

$\left[\left\{5\right\}\&comma;\left\{1\&comma;2\right\}\&comma;\left\{2\&comma;3\right\}\&comma;\left\{1\&comma;4\right\}\&comma;\left\{2\&comma;4\right\}\right]$

$T≔\mathrm{Transversal}\left(R\right)\&semi;$$\mathrm{Hyperedges}\left(T\right)$

$T≔\mathrm{<\; a\; hypergraph\; on\; 5\; vertices\; with\; 3\; hyperedges\; >}$

$\left[\left\{1\&comma;2\&comma;5\right\}\&comma;\left\{2\&comma;4\&comma;5\right\}\&comma;\left\{1\&comma;3\&comma;4\&comma;5\right\}\right]$

$\mathrm{TT}≔\mathrm{Transversal}\left(T\right)\&semi;$$\mathrm{Hyperedges}\left(\mathrm{TT}\right)$

$\mathrm{TT}≔\mathrm{<\; a\; hypergraph\; on\; 5\; vertices\; with\; 5\; hyperedges\; >}$

$\left[\left\{5\right\}\&comma;\left\{1\&comma;2\right\}\&comma;\left\{2\&comma;3\right\}\&comma;\left\{1\&comma;4\right\}\&comma;\left\{2\&comma;4\right\}\right]$

$\mathrm{AreEqual}\left(\mathrm{TT}\&comma;M\right)$

$\mathrm{true}$

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.

The Hypergraphs[Max] command was introduced in Maple 2024.

For more information on Maple 2024 changes, see Updates in Maple 2024.