I came across following fact while reading online:
If a given Hasse diagram contains following structures, then it wont be the distributive lattice
I am unsure if the above fact is correct and if it is, then whether are these exhaustive substructures required to be contained in given Hasse diagram in order for a lattice not a distributive lattice. In other words, if the fact is true, then can there be a non distributive lattice with Hasse diagram not containing any of these structures?
From Wikipedia's article on distributive lattices: "A lattice is distributive if and only if none of its sublattices is isomorphic to $M_3$ or $N_5$."
Yes, it is correct. It is a complete characterization of distributive lattices.
The proof is not difficult, but it is rather elaborated.