The ideal of Plücker relations has a quadratic Groebner basis. Under a suitable term order, the incomparable pairs of the poset $P$ generate the initial ideal of the ideal of Plücker relations.
Given two subsets $S$ and $T$ of ${0,\ldots,n-1}$, we partially order $S \leq T$ if $\#S \geq \#T$ and $S_i \leq T_i$ for all $i$ from $1$ to $\#T$.
i1 : P = plueckerPoset 4; |
i2 : coveringRelations P
o2 = {{{0}, {1}}, {{1}, {2}}, {{0, 1}, {0, 2}}, {{2}, {3}}, {{0, 2}, {0, 3}},
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{{0, 2}, {1, 2}}, {{1, 2}, {1, 3}}, {{0, 1, 2}, {0, 1, 3}}, {{3}, {}},
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{{0, 3}, {0}}, {{0, 3}, {1, 3}}, {{1, 3}, {2, 3}}, {{1, 3}, {1}}, {{0,
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1, 3}, {0, 2, 3}}, {{0, 1, 3}, {0, 1}}, {{2, 3}, {2}}, {{0, 2, 3}, {0,
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2}}, {{0, 2, 3}, {1, 2, 3}}, {{1, 2, 3}, {1, 2}}, {{0, 1, 2, 3}, {0, 1,
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2}}}
o2 : List
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The object plueckerPoset is a method function.