The product of two lattices using the lexical order may not be a lattice at all.
Suppose that L and K are lattices, and that L is not a linear order and that K has no least element. 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.