The command TransitiveReduction(P) returns as an adjacency matrix the transitive reduction of the partially ordered set P.

$\mathrm{with}\left(\mathrm{PartiallyOrderedSets}\right)\:$

$S≔\left\{1\&comma;2\&comma;3\&comma;4\&comma;5\right\}\&colon;$$\mathrm{leq}≔\mathrm{`<=`}\&colon;$$\mathrm{poset1}≔\mathrm{PartiallyOrderedSet}\left(S\&comma;\mathrm{leq}\right)$

$\mathrm{poset1}≔\mathrm{<\; a\; poset\; with\; 5\; elements\; >}$

$\mathrm{DrawGraph}\left(\mathrm{poset1}\right)$

$\mathrm{TransitiveReduction}\left(\mathrm{poset1}\right)$

$\left[\begin{array}{ccccc}1& 1& 0& 0& 0\\ 0& 1& 1& 0& 0\\ 0& 0& 1& 1& 0\\ 0& 0& 0& 1& 1\\ 0& 0& 0& 0& 1\end{array}\right]$

$\mathrm{lneq}≔\mathrm{`<`}\&colon;$$\mathrm{poset1\_1}≔\mathrm{PartiallyOrderedSet}\left(S\&comma;\mathrm{lneq}\right)$

$\mathrm{poset1\_1}≔\mathrm{<\; a\; poset\; with\; 5\; elements\; >}$

$\mathrm{DrawGraph}\left(\mathrm{poset1\_1}\right)$

$\mathrm{TransitiveReduction}\left(\mathrm{poset1\_1}\right)$

$\left[\begin{array}{ccccc}1& 1& 0& 0& 0\\ 0& 1& 1& 0& 0\\ 0& 0& 1& 1& 0\\ 0& 0& 0& 1& 1\\ 0& 0& 0& 0& 1\end{array}\right]$

$\mathrm{divisibility}≔\left(x\&comma;y\right)\mapsto \mathrm{irem}\left(y\&comma;x\right)=0\&colon;$$T≔\left\{3\&comma;4\&comma;5\&comma;6\&comma;7\&comma;8\&comma;9\right\}\&colon;$

$\mathrm{poset2}≔\mathrm{PartiallyOrderedSet}\left(T\&comma;\mathrm{divisibility}\right)$

$\mathrm{poset2}≔\mathrm{<\; a\; poset\; with\; 7\; elements\; >}$

$\mathrm{DrawGraph}\left(\mathrm{poset2}\right)$

$\mathrm{TransitiveReduction}\left(\mathrm{poset2}\right)$

$\left[\begin{array}{ccccccc}1& 0& 0& 1& 0& 0& 1\\ 0& 1& 0& 0& 0& 1& 0\\ 0& 0& 1& 0& 0& 0& 0\\ 0& 0& 0& 1& 0& 0& 0\\ 0& 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 0& 1\end{array}\right]$

$\mathrm{divisibNE}≔\left(x\&comma;y\right)\mapsto \mathrm{irem}\left(y\&comma;x\right)=0\mathbf{and}y\ne x\&colon;$

$\mathrm{poset2\_1}≔\mathrm{PartiallyOrderedSet}\left(T\&comma;\mathrm{divisibNE}\&comma;\mathrm{reflexive}=\mathrm{checkfalse}\right)$

$\mathrm{poset2\_1}≔\mathrm{<\; a\; poset\; with\; 7\; elements\; >}$

$\mathrm{DrawGraph}\left(\mathrm{poset2\_1}\right)$

$\mathrm{TransitiveReduction}\left(\mathrm{poset2\_1}\right)$

$\left[\begin{array}{ccccccc}1& 0& 0& 1& 0& 0& 1\\ 0& 1& 0& 0& 0& 1& 0\\ 0& 0& 1& 0& 0& 0& 0\\ 0& 0& 0& 1& 0& 0& 0\\ 0& 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 0& 1\end{array}\right]$

Richard P. Stanley: Enumerative Combinatorics 1. 1997, Cambridge Studies in Advanced Mathematics. Vol. 49. Cambridge University Press.

The PartiallyOrderedSets[TransitiveReduction] command was introduced in Maple 2025.

For more information on Maple 2025 changes, see Updates in Maple 2025.