Let $S$ be a set of all partitions defined on a nonempty set $A$. The reation $R$ on a set $S$ is defined to be $\langle\pi_1, \pi_2\rangle \in R$ iff $\pi_1$ refines $\pi_2$.
(a) show that $R$ is a partialreally struggel ordering.
This I had no problems with and simply showed that R is antisymmetric, transitive, and reflexive.
(b) Is the poset $\langle S,R\rangle$ a lattice?
I am really struggling with this one. I know a lattice means that every pair in S has a least upper bound (lub) and a greatest lower bound (glb). I can't really think of a counter example because I cannot stop thinking of $\langle \pi_1, \pi_2\rangle$ in abstract terms.
Am I right to assume that $\langle S,R\rangle = { \pi_1 \leq \pi_2 \leq \pi_3 .... \leq \pi_n} $
Now every pair in S, am I referring to $\langle \pi_n, \pi_m\rangle$ for any $n$, $m$?
I guess I need to come up with a counter example here, since my set is not really defined but in abstract terms. I don't quite see how and why?
I think I haven't fully understood what a lub, and glb (join and meet) are in the sense of a lattice. When I look at examples I cannot understand it and I think I am taking it too literally. Could someone provide me with a detailed example showing lub and glb for a certain set as a lub, and glb and explain why?
I would appreciate any help and corrections in the error of my thinking!!