isLowerSemilattice -- determines if a poset is a lower (or meet) semilattice

Synopsis

The poset $P$ is a lower semilattice if every pair of vertices has a unique greatest lower bound (meet).

Clearly, the $n$ chain and the $n$ booleanLattice are lower semilattices.

i1 : n = 4;

i2 : isLowerSemilattice chain n

o2 = true

i3 : B = booleanLattice n;

i4 : isLowerSemilattice B

o4 = true

The middle ranks of the $n$ booleanLattice are not lower semilattices.

i5 : isLowerSemilattice flagPoset(B, {1,2,3})

o5 = false

However, the lower ranks of the $n$ booleanLattice are non-lattice lower semilattices.

i6 : B' = flagPoset(B, {0,1,2,3});

i7 : isLattice B'

o7 = false

i8 : isLowerSemilattice B'

o8 = true