I have just began reading through Steven Roman's "Lattices and ordered sets", and I came across an exercise in Chapter 1 that I can't seem to find a good answer to. All the others are fairly easy, so either I'm missing something trivial or maybe I misunderstand the question. The exercise is as following:
"Find a poset $P$ that contains an antichain $A=\{x,y,z\}$ with the property that no pair of elements from $A$ has a join but $A$ has a join." (A join of a set is the same as the least upper bound for that set.)
I'm struggling with the question, perhaps because of some ambiguity. To illustrate: suppose $P = \{\{1,2\},\{2,3\},\{1,3\},\{1,2,3\}\}$ with $\leq$ being 'is a subset of' and the antichain $A = \{\{1,2\},\{2,3\},\{1,3\}\}$ then $\{1,2,3\}$ is a join for $A$, but doesn't this imply that it is a join for $\{\{1,2\},\{2,3\}\}$ as well? Although it is join for more than just this pair of elements, but still a valid join I'd say?
So concluding from this, for finite $P$, if the antichain $A$ has a join $j$ then $j$ would be an upper bound for each of the pairs $\{\{x,y\},\{x,z\},\{y,z\}\}$ as well and so they definitely have a join, no? So that leaves just the infinite posets to think about, and I haven't really found an example that fulfills the criteria of the exercise.
Hint: Think about the natural numbers ordered by divisibility. For simplicity choose $x,y,z$ all different prime numbers and remove from $\mathbb N$ the joins of each pair of elements of $A$.