Lesson 1

Small partially ordered sets can be conveniently represented as Hasse diagrams: draw an edge from element \(x\) to element \(y\) if \(x \prec y\) and there is no element \(z\) such that \(x \prec z \prec y\). Moreover, let us agree to place an element at level \(k\) (levels are numbered bottom-up) if the longest chain ending at this element consists of \(k\) elements. It is easy to see that any horizontal slice is an antichain (but there may be other antichains!), and any path is a chain (again, there may be other chains).

Here is a Hasse diagram for subsets of \(\{a,b,c\}\) with the inclusion relation:

And here is the Hasse diagram for all non-unit digits with the divisibility relation (assume \(0 \preceq 0\)).