I call a system of precedences $U$ a set of nonempty subsets of some poset. I will denote $A<B \Leftrightarrow \forall a\in A, b\in B: a<b$ for sets $A,B\in U$.
Find sufficient and necessary restrictions on binary relations $<$ and $\subseteq$ such that there exists a system $U$ of precedences such that they are exactly (up to isomorphism) $<$ and $\subseteq$ on $U$.
(I wrote my conditions for these operations for a system of precedences in https://cs.stackexchange.com/q/85951/39512 but I'm not sure if these conditions are right.)
The proposed conditions in that answer can be written as the following:
$\subseteq$ is a non-strict partial order relation and $<$ is a strict partial order relation.
$\forall a,b,a_1,b_1\in U:(a<b \wedge a_1\subseteq a \wedge b_1\subseteq b \Rightarrow a_1<b_1)$.
(Not relevant to the question, just where the questions appeared from) Systems of precedences origin from subsets of $U$ being sets of operations, where operations with higher precedences should be applies before operations of lower precedences.)
You may assume that all sets in consideration are finite.