The command LeastElement(P) returns the least element of the partially ordered set P, if this element exists, otherwise NULL is returned.

If noleastelement = s is passed as an optional argument, and if P does not have a greatest element, then s is returned

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

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

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

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

$\mathrm{LeastElement}\left(\mathrm{empty\_poset}\&comma;\mathrm{noleastelement}=''no\; least\; element\; found''\right)$

$''no\; least\; element\; found''$

$\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;$

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

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

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

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

$\mathrm{LeastElement}\left(\mathrm{poset2}\&comma;\mathrm{noleastelement}=''no\; least\; element\; found''\right)$

$''no\; least\; element\; found''$

$Z≔\left\{1\&comma;2\&comma;3\&comma;4\&comma;5\&comma;6\&comma;10\&comma;12\&comma;15\&comma;20\&comma;30\&comma;60\right\}$

$Z≔\left\{1\&comma;2\&comma;3\&comma;4\&comma;5\&comma;6\&comma;10\&comma;12\&comma;15\&comma;20\&comma;30\&comma;60\right\}$

$\mathrm{poset10}≔\mathrm{PartiallyOrderedSet}\left(Z\&comma;\mathrm{divisibility}\right)$

$\mathrm{poset10}≔\mathrm{<\; a\; poset\; with\; 12\; elements\; >}$

$\mathrm{DrawGraph}\left(\mathrm{poset10}\right)$

$\mathrm{LeastElement}\left(\mathrm{poset10}\right)$

$1$

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

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

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