Consider this Hasse Diagram of a poset:
This poset is not a complete Lattice.
From definition of complete lattice, we need to have a least upper bound and a greatest lower bound for each pair of elements (is this right)?
From what I understood, this is not a complete Lattice because we have for example no lower bound for (a,b).
But my question is whether in this poset we have a least upper bound for (a,b).
The candidates are $c,d,e$ but there is no least one because $$ c \leq e \quad d\leq e \quad c \nleq d \quad d \nleq c $$
Or can we just choose c or d as our least upper bound?
EDIT:
So, in the figure above, we don't have a lattice, since there is no least upper bound for (a,b), and there is no greater lower bound for (c,d).
So now I decided to change the diagram in this way:
In this case can we say that it is a complete lattice? Can we correctly say that we can choose c or d to be our least upper bound for (a,b). And choose a or b to be our greatest lower bound for (c,d)?
Among the upper bounds, there is no least element as you correctly pointed out. A least element of a set is an element that is less than any other element in the set (so no, we can't just choose $c$ or $d$, they aren't the least)
Your definition of complete lattice is slightly wrong tough. You gave de definition for a lattice: every pair of elements has a least upper bound and greatest lower bound). For a complete lattice you demand that every subset has a least upper bound and greatest lower bound. This is a stronger condition.
So in the beginning of your post you actually showed that your poset isn't even a lattice. (let alone a complete lattice)