If a moment graph $G$ arises from a (possibly singular) GKM variety $X$ with an equivariant stratification, with each strata having a unique torus-fixed point, the vertices of $G$ (which correspond to the torus-fixed point of $X$) form a poset where $v_1 \leq v_2$ if the closure of the stratum corresponding to $v_1$ contains that of $v_2$. The following example features the Schubert variety of projective lines in $\mathbb P^3$ meeting a distinguished line. The poset of its stratification by smaller Schubert cells is a subposet of the Bruhat poset.
i1 : Gr24 = generalizedFlagVariety("A",3,{2})
o1 = a GKM variety with an action of a 4-dimensional torus
o1 : GKMVariety
|
i2 : X = generalizedSchubertVariety(Gr24, {set{0,2}})
o2 = a GKM variety with an action of a 4-dimensional torus
o2 : GKMVariety
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i3 : cellOrder X
o3 = Relation Matrix: | 1 1 1 1 1 |
| 0 1 0 1 1 |
| 0 0 1 1 1 |
| 0 0 0 1 1 |
| 0 0 0 0 1 |
o3 : Poset
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The object cellOrder is a method function.