This screenshot is from Invitation to Discrete Mathematics by Matoušek. :
I understand the definition when two posets (or ordered sets) are isomorphic or non-isomorphic. However, I am confused about question (a), the selected line in the screenshot.
More generally, when we talk about Non-Isomorphic n-element posets, do we mean that they are pair-wise non-isomorphic, or do we mean that for any of these posets, we cannot find another poset that is isomorphic to it (which would imply the previous I guess)?
Thanks!
You have to draw $n$ many Hasse diagrams for posets $P_1, P_2, \ldots P_n$, all having $3$ elements, such that if $i \neq j$ then $P_i$ is not isomorphic to $P_j$ and moreover if $P$ is any three-element poset there is an $i\in \{1,2,\ldots,n\}$ so that $P$ is isomorphic to $P_i$.
So you have then a "catalogue" of all "types" of posets of that size with a unique representative for all. Your job is to figure out $n$ and the $P_i$.