Give an example of a subset $L'$ of a lattice $L$ , such that $L'$ is a lattice but not a sublattice of $L$.

Question

Find the example of a subset $L'$ of a lattice $L$ , such that $L'$ is a lattice but not a sublattice of $L$.

Answer 1

Take a 2x2 lattice with elements 0, a, b, c. Add a new top, 1. Now delete c. The resulting subset is a lattice. If it were a sublattice, then the join of a and b would have to be c (but it's 1).

The point is, sublattices are subsets that are closed under the (over)lattice's operations.

Answer 2

You can order-embed either the pentagon $\mathbf{N}_5$ or the diamond $\mathbf{M}_3$ in the 8 element boolean lattice, the cube $\mathbf{2}^3$.
However, this one is distributive and those are not, so they're not sublattices of the cube.