isUpperSemimodular -- determines if a lattice is upper semimodular

Synopsis

Usage:

i = isUpperSemimodular P
Inputs:
- P, an instance of the type Poset, a ranked lattice
Outputs:
- i, a Boolean value, whether $P$ is upper semimodular

Description

Let $r$ be the ranking of $P$. Then $P$ is upper semimodular if for every pair of vertices $a$ and $b$, $r(a) + r(b) \geq r(join(a,b)) + r(meet(a,b,))$.

The $n$ chain and the $n$ booleanLattice are upper semimodular.

i1 : n = 4;

i2 : isUpperSemimodular chain n

o2 = true

i3 : isUpperSemimodular booleanLattice n

o3 = true

The following lattice is not upper semimodular.

i4 : P = poset {{1, 2}, {1, 3}, {1, 4}, {2, 5}, {2, 6}, {3, 5}, {4, 6}, {5, 7}, {6, 7}};

i5 : isLattice P

o5 = true

i6 : isUpperSemimodular 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.

Ways to use isUpperSemimodular :

isUpperSemimodular(Poset)

For the programmer

The object isUpperSemimodular is a method function.

isUpperSemimodular -- determines if a lattice is upper semimodular

Synopsis

Description

See also

Ways to use isUpperSemimodular :

For the programmer