whether given lattice is distributive or complemented or both?

Question

whether given lattice is distributive or complemented or both?

       a
      /|\
     / | \
    b  |  c
    |  d  |
    e  |  f
    \  | / 
     \ |/
       g

For a lattice to be distributive each element should have unique complement. here d can have two b & c , hence it is not distributive. Am I right?

Answer 1

You are correct. Another way to see this (which might be easier for more complicated lattices) is to note that the lattice has $N_5$ and $M_3$ as sublattices. A lattice is distributive iff it has no sublattice isomorphic to $N_5$ or $M_3$. (A lattice is modular iff it has no sublattice $\cong N_5$.)