Lattice which is not bounded lattice

Question

I want to find an example of a lattice which is not a bounded lattice . Diagrams would be good with an explanation .

Answer 1

All finite lattices are bounded. $(\Bbb N,|)$ is an example of an unbounded lattice. Or you can add the Hasse diagram of $\Bbb Z$ above the maximum element of the Hasse diagram of any bounded lattice to get an unbounded lattice.

Answer 2

The given poset is a lattice because any given pair of elements do have a lowest upper bound (LUB), but it tends to positive infinite for very large integers. By a similar reasoning, the GLB tends to negative infinite for very large negative integers. Therefore, LUB and GLB, even though they exist, are not bounded. Thus, the given poset is an unbounded lattice. This concept would be clearer if you were to draw the same conclusions from the Hasse diagram.