The command AreEqual(P1, P2) checks whether the partially ordered sets P1 and P2 have the same elements and the same relations. To be more precise, let us denote by V the underlying set of P1, by R1 (resp. R2 ) the binary relation on V defining P1 (resp. P2). The posets P1 and P2 are equal whenever V is also the underlying set of P2 and for any two elements a and b in V,  R1(a,b) holds if and only if R2(a,b) holds.

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

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

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

$\mathrm{AreEqual}\left(\mathrm{empty\_poset}\&comma;\mathrm{empty\_poset}\right)$

$\mathrm{true}$

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

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

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

$\mathrm{AreEqual}\left(\mathrm{empty\_poset}\&comma;\mathrm{poset1}\right)$

$\mathrm{false}$

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

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

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

$\mathrm{AreEqual}\left(\mathrm{poset1}\&comma;\mathrm{poset1\_1}\right)$

$\mathrm{true}$

$\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;$

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

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

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

$\mathrm{AreEqual}\left(\mathrm{poset1}\&comma;\mathrm{poset2}\right)$

$\mathrm{false}$

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

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

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