Fix an (infinite) set $U$.
Denote $F$ the set of filters on $U$.
An isotone family of filters is a set $I\subseteq F$ such that $A\supseteq B\implies A\in I$ for every $B\in I$ and filter $A$ (on $U$).
Let us order the set $L$ of all isotone families of filters (on $U$) by set inclusion.
Problem Is there a poset $T$ such that $L$ is order-isomorphic to a. filters on $T$; b. ideals on $T$?
If yes, can we add additional requirements for $T$ (e.g. require that $T$ is a lattice, a distributive lattice, a boolean lattice, a complete lattice, etc.?)
Remark: I consider improper filter as a filter, for completeness.