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Posets :: isModular

isModular -- determines if a lattice is modular

Synopsis

Description

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.

See also

Ways to use isModular :