The Hasse diagrams of all the lattices with up to four elements are ________ (write all the relevant Hasse diagrams).

GATE 1994 · Discrete Mathematics · Lattice · medium

Answer: There are 5 non-isomorphic lattices with up to four elements: the 1-point lattice, the 2-, 3- and 4-element chains, and the 4-element diamond (2x2 square).

1. Sizes 1, 2 and 3: size 1: the single point is a lattice (1). size 2: the only poset that is a lattice is the chain 0 < 1 (1). size 3: a chain 0 < m < 1 is a lattice, but the 'V' (one bottom, two incomparable tops) has no join for the two tops, and the inverted 'V' has no meet for the two bottoms, so only the chain qualifies (1).
2. Size 4: keep only valid lattices: the 4-element chain 0 < a < b < 1 is a lattice. The diamond (bottom 0, top 1, two incomparable middles a and b) is a lattice: a join b = 1, a meet b = 0. Other 4-element posets (the 'N', the 'V with a tail', two incomparable tops, etc.) fail the lattice test, so size 4 contributes 2 lattices.
3. Add the counts: sum the lattices found for sizes 1, 2, 3, 4: 1 (point) + 1 (2-chain) + 1 (3-chain) + 2 (4-chain and diamond) = 5.