In each case, we need to check whether every pair of elements has both a least upper bound and a greatest lower bound.
a) This is a lattice. If we want to find the l.u.b. or g.l.b. of two elements in the same vertical column of the Hasse diagram, then we simply take the higher or lower (respectively) element.
If the elements are in different columns, then to find the g.l.b. we follow the diagonal line upward from the element on the left, and then continue upward on the right, if necessary to reach the element on the right.
For example, the l.u.b. of d and c is f; and the l.u.b. of a and e is e.
Finding greatest lower bounds in this poset is similar.
b) This is not a lattice. Elements b and c have f, g, and h as upper bounds, but none of them is a l.u.b.
c) This is a lattice. By considering all the pairs of elements, we can verify that every pair of them has a l.u.b. and a g.l.b.
For example, b and e have g and a filling these roles, respectively.
