The command ConnectedComponents(P) returns as a set of sets the connected components of the partially ordered set P.

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

$V≔\varnothing \&colon;$$\mathrm{leq}≔\mathrm{`<=`}\&colon;$$\mathrm{empty\_poset}≔\mathrm{PartiallyOrderedSet}\left(V\&comma;\mathrm{leq}\right)$

$\mathrm{empty\_poset}≔\mathrm{<\; a\; poset\; with\; 0\; elements\; >}$

$\mathrm{ConnectedComponents}\left(\mathrm{empty\_poset}\right)$

$\varnothing$

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

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

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

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

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

$\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;$

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

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

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

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

$\left\{\left\{5\right\}\&comma;\left\{7\right\}\&comma;\left\{4\&comma;8\right\}\&comma;\left\{3\&comma;6\&comma;9\right\}\right\}$

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

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

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