Let $r$ be the ranking of $P$. Then $P$ is modular if for every pair of vertices $a$ and $b$, $r(a) + r(b) = r(join(a,b)) + r(meet(a,b,))$. That is, $P$ is modular if it isLowerSemimodular and isUpperSemimodular.
The $n$ chain and the $n$ booleanLattice are modular.
i1 : n = 4; |
i2 : isModular chain n o2 = true |
i3 : isModular booleanLattice n o3 = true |
The following lattice is not modular.
i4 : P = poset {{1, 2}, {1, 5}, {2, 3}, {2, 4}, {3, 7}, {4, 7}, {5, 4}, {5, 6}, {6, 7}};
|
i5 : isLattice P o5 = true |
i6 : isModular P o6 = false |
This method uses the methods isLowerSemimodular and isUpperSemimodular, which were ported from John Stembridge's Maple package available at http://www.math.lsa.umich.edu/~jrs/maple.html#posets.
The object isModular is a method function.