What's wrong with just adding a top and bottom element to a poset to convert it into a lattice?

Question

I'm a novice in order theory. This question (Smallest lattice containing a poset) and its answers indicate that it is not so straightforward. And I want to know where my reasoning is going wrong.

Following are my steps to convert a poset to a lattice. I have not been able to prove them correct but unfortunately I haven't found a counter-example either.

1. To a set with a partial order $(S, \leq)$ add two special elements $\top$ and $\bot$ to make $S'$ and extend $\leq$ to make $\leq'$ such that $\forall s \in S,~ \bot \leq' s \leq' \top$
2. Define the meet operation $a \land b$ as the least common ancestor of a and b in a graph where $\leq'$ forms directed edges, and the join operation $a \lor b$ as the first common descendant. Both of these exist (due to $\top$ and $\bot$) and so these operations are well defined.

Now $(S', \lor, \land)$ form a lattice.

Is there a counter-example to help me see that this definition would lead to violation of one of the laws for join and meet operations as I've defined them? Or is it the case that this is actually a valid lattice?

Answer 1

Because just adding a greatest and a least element doesn't fix pairs that have upper (or lower) bounds, but no least upper (greatest lower) bound.

For instance, take the set $\{a,b,c,d\}$ with partial order given by $$ a<c\\ b<c\\ a<d\\ b<d $$ Then the subset $\{a,b\}$ is bounded above; both $c$ and $d$ are upper bounds. However, $c$ and $d$ are incomparable, so there is no least upper bound.

Adding a new element that is larger than both $c$ and $d$, and a new element that is smaller than both $a$ and $b$, doesn't remedy this. You would also need to insert an element "between" $\{a,b\}$ and $\{c,d\}$. Inserting such elements for a general poset (rather than perhaps the simplest example there is as I've done here), and at the same time proving that there is a minimal way of doing so, is not an easy task.

Answer 2

I haven't checked the following, but this is more in line of a comment than an answer but as my phone isn't allowing me to add comments, I'm just adding it here.

I think you've shown how to complete any poset into a lattice. In terms of category theory there's an inclusion functor of Lat into Pos. This has a left adjoint, called the reflector, and this adds top and bottom to a given poset in the way that you've indicated. We also say, that Lat is a reflective subcategory of Pos.