A cardinal $\kappa$ is huge iff there is $\lambda>\kappa$ and a $\kappa$-complete normal ultrafilter on
$P\_{\leq \kappa}(\lambda)$, or, equivalently, on the set of families of subsets of $\lambda$ of order-type $\kappa$. [1]
I want to know whether one can further assume that $U$ is concentrated on something smaller? For example, can one further assume that $U$ is concentrated on $<\kappa$-directed families of sets of size $\kappa$?
Only on those directed families that contain $P\_{\kappa}(\lambda)$? Covering families as above ?


In notation: Let $U$ be such an ultrafilter. Let $D\_\kappa=\{X\in P\_{\leq\kappa}(\lambda): \forall S\subseteq X(|S|<\kappa\implies \cup S \in X\} $ be the subset of all $\kappa$-directed subsets of $P\_{\leq \kappa}(\lambda)$.
Is $D\_\kappa\in U $ ? Let $D'\_\kappa=\{X\in D\_\kappa: P\_{<\kappa}(\lambda)\subseteq X\}$ be the subset of all
$\kappa$-directed subsets of $P\_{\leq \kappa}(\lambda)$ containing all sets of size less than $\kappa$.
Let $C'\_\kappa=\{X\in D\_\kappa: \forall S\in P\_{\leq\kappa}(\lambda)\exists x\in X(S\subseteq x)\}$.  

Is $D'\_\kappa\in U$ ? Is $C'\_\kappa\in U$? Can one find such an $U$ that the answers are positive? 


I am also looking for any references describing in detail ultrafilter characterizations of large cardinals. 


[1] The Ultrafilter Characterization of Huge Cardinals Robert J. Mignone Proceedings of the American Mathematical Society Vol. 90, No. 4 (Apr., 1984), pp. 585-590 <http://www.ams.org/proc/1984-090-04/S0002-9939-1984-0733411-6/S0002-9939-1984-0733411-6.pdf>



ultrafilter characterisation of huge cardinals
The hugeness of $\kappa$ is witnessed by an embedding $j:V\to M$ for which $M^\lambda\subset M$, where $\lambda=j(\kappa)$. In particular, for such an embedding we have $j''\lambda\in M$, and one may accordingly consider the induced measure on $P\_{\leq\kappa}(\lambda)$ defined by $X\in U\iff j''\lambda\in j(X)$. The ultrapower by $U$ is a factor of the original embedding $j$ and also witnesses the hugeness of $\kappa$. 


Since $j''\lambda$ is $\lt\lambda$-closed, in the sense that every subset of it of size less than $\lambda$ is bounded in it, then we see that $U$ concentrates on such kind of objects. This kind of reasoning is usually the key for understanding the essential nature of the sets on which a hugeness measure must concentrate.


But I am confused about your question concerning $\kappa$-directedness, since it seems to me that every subset $\sigma\subset\lambda$ with order type $\kappa$ will be $\kappa$-directed, since any small subset of $\sigma$ will be bounded in $\sigma$. Have you asked the question you intended to ask?


In your comment you state another property, which I would call something like $\lt\kappa$-closure rather than $\kappa$-directedness, since it refers to the least upper bound $\cup S$ rather than merely to an upper bound of $S$. In this case, $j''\lambda$ is not $\lt\lambda$-closed, since the union of the first $\kappa$ many elements of it is $\kappa$ itself, which is not an element of $j''\lambda$. So the usual hugeness measures will not concentrate on $D\_\kappa$ defined that way.


