The command LeastUpperBound(P,E1,E2) returns the least upper bound of the pair consisting of the elements E1 and E2 in the poset P, if this least upper bound exists, otherwise NULL is returned.

The command LeastUpperBound(P,L) returns the least upper bound of the subset L of the underlying set of P, if this least upper bound exists, otherwise NULL is returned.

If nosharedupperbounds = s  (resp. multiplesharedupperbounds = s) is provided, and if no least upper bound exists because no shared upper bounds exist (resp. because multiple shared upper bounds exist) then s is returned.

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

$\mathrm{leq}≔\mathrm{`<=`}\&colon;$

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

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

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

$\mathrm{LeastUpperBound}\left(\mathrm{poset1}\&comma;3\&comma;4\right)$

$4$

$\mathrm{LeastUpperBound}\left(\mathrm{poset1}\&comma;\left\{3\&comma;4\right\}\right)$

$4$

$\mathrm{LeastUpperBound}\left(\mathrm{poset1}\&comma;\left\{3\&comma;4\&comma;5\right\}\right)$

$5$

$\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;$

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

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

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

$\mathrm{LeastUpperBound}\left(\mathrm{poset2}\&comma;5\&comma;7\right)$

$\mathrm{LeastUpperBound}\left(\mathrm{poset2}\&comma;5\&comma;7\&comma;\mathrm{nosharedupperbounds}=''no\; shared\; upper\; bounds''\right)$

$''no\; shared\; upper\; bounds''$

$\mathrm{LeastUpperBound}\left(\mathrm{poset2}\&comma;\left\{3\&comma;9\right\}\right)$

$9$

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

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

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