Discussion
The number of non-isomorphic modular lattices on 5 elements is :
Answer: B
There are 4 non-isomorphic modular lattices on 5 elements.
There are exactly five lattices with 5 elements: the chain; the diamond \(M_{3}\) (a top, a bottom and three incomparable atoms); the pentagon \(N_{5}\); and the two lattices formed from the four-element Boolean square by adding a new top or a new bottom.
A lattice is modular unless it contains a copy of \(N_{5}\), so the pentagon is the only non-modular one. The chain and the two square-based lattices are even distributive, and \(M_{3}\) is modular but not distributive. That leaves 4 modular lattices.
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