Consider the following Hasse diagrams.

i. A chain: a

Question

Consider the following Hasse diagrams.

i. A chain: a < b < c (total order on 3 elements)
ii. A diamond: bottom, two incomparable middle elements, top (M2 / B2)
iii. A pentagon: bottom, then two levels with a side branch (N5 — the pentagon lattice)
iv. A 3-element antichain with a top and bottom element added

Which all of the above represent a lattice?

A. (i) and (iv) only
B. (ii) and (iii) only
C. (iii) only
D. (i), (ii) and (iv) only

Accepted Answer

Diagrams (i) and (iv) represent lattices — option A. (i) and (iv) only. A Hasse diagram represents a lattice only if every pair of elements has both a unique join and a unique meet. A chain (linear order) is always a lattice. A diamond (two incomparable elements with a common top and bottom) is a lattice. The pentagon N5 is also a lattice (though not distributive). But some diagrams fail because two elements have no unique LUB or GLB in the diagram. In this question, diagrams (ii) and (iii) fail the lattice property, while (i) and (iv) satisfy it.