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

The command GreatestLowerBound(P,L) returns the greatest lower bound of the subset L of the underlying set of P, if this greatest lower bound exists, otherwise NULL is returned.

If nosharedlowerbounds = s  (resp.  multiplesharedlowerbounds = s) is provided, and if no greatest lower bound exists because no shared lower bounds exist (resp. because multiple shared lower 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)$

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

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

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

$3$

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

$3$

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

$3$

$\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;$

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

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

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

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

$\mathrm{GreatestLowerBound}\left(\mathrm{poset2}\&comma;5\&comma;7\&comma;\mathrm{nosharedlowerbounds}=''no\; shared\; lower\; bounds''\right)$

$''no\; shared\; lower\; bounds''$

$\mathrm{GreatestLowerBound}\left(\mathrm{poset2}\&comma;\left\{6\&comma;9\right\}\right)$

$3$

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

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

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