Let $r$ be the ranking of $P$. Then $P$ is lower semimodular if for every pair of vertices $a$ and $b$, $r(a) + r(b) \leq r(join(a,b)) + r(meet(a,b,))$.
The $n$ chain and the $n$ booleanLattice are lower semimodular.
i1 : n = 4; |
i2 : isLowerSemimodular chain n o2 = true |
i3 : isLowerSemimodular booleanLattice n o3 = true |
The following lattice is not lower semimodular.
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 : isLowerSemimodular P o6 = false |
This method was ported from John Stembridge's Maple package available at http://www.math.lsa.umich.edu/~jrs/maple.html#posets.
The object isLowerSemimodular is a method function.