The usual order to put on the product of two lattices is the product order, rather than the lexical order. With the product order, one should simply take lub or glb separately in each coordinate, and it is not difficult to see that this is a latice order. 

If one uses the lexical order, however, then the product of two lattices may not be a lattice at all.

To see this, suppose that L and K are lattices, such that L is not a linear order and K has no least element, then the lexical order on L x K is not a lattice order. Let a and b be elements of L that are incomparable, and let c be any element of L. I claim that (a,c) and (b,c) have no least upper bound. To see this, suppose that (x,y) is the least upper bound of (a,c) and (b,c). It must be that a and b are both ≤ x, and so lub(a,b) ≤ x. Since a and b are incomparable, they are both strictly less than lub(a,b). Thus, any element of the form (lub(a,b),y) is an upper bound. But there is no least such element, since L has no least element. 

However, if the lattices is complete, then the lexical order is a lattice order.