How many non-isomorphic partial orders of a certain size?

Question

I'm working on the consistency of Martin's axiom and I need some help counting. Assume we are in a universe where GCH holds and $\kappa$ is a regular cardinal. How many non-isomorphic partial orders are there of size less than $\kappa$. I think the answer is $\kappa$ (indeed this is what Jech writes) and can convince myself, but it is not very clear to me how to systematically count the number of different partial orders of a certain size.

Let me know if I need to give more context, I'm asking my specific question in a general way.

Answer 1

Suppose $(P,\le)$ is a partial order and $|P|<\kappa$. We can embed every such $P$ in its power set, using $p\in P$ then $p\mapsto\{x\in P\mid x\le p\}$. If we assume GCH then without loss of generality $P=\lambda<\kappa$ then $P(\lambda)=\lambda^+\le\kappa$.

Now we have that every partial order of cardinality less than $\kappa$ can be embedded into its power set, which is at most $\kappa$.

Note that every possible partially ordered set can be embedded this way into its power set, thus:

1. If $\kappa$ is a limit cardinal (and of course regular) every power set is smaller, the number of partially ordered sets is at most $\lambda^+$ for $\lambda<\kappa$, and thus we have $\kappa^{<\kappa}=\kappa$ many partially ordered sets of size $<\kappa$.
2. If $\kappa=\mu^+$ is a successor cardinal (thus regular assuming choice) then every partially ordered set can be embedded in $P(\mu)=\kappa$ - by GCH - and so there are only $\kappa$ many of those.