I need help finding an appropriate name for the following object.
Consider a set (call it unit set) A. Let there be some set K, which is a subset of the power set of A, $K \subseteq 2^A = \{K_1,K_2,K_3...K_n\}$, so that each $ K_i $ is a subset of A for all $ i$.
The set K is said to be "admissible" if the following holds: $K_1, K_2 \in K$, $K_1 \cap K_2 \ne \emptyset$ implies $K_1 \cup K_2 \in K$.
So, it is as if $K$ is "closed under a non-disjoint union".
The question is - does such an object/structure or the family of such sets have a name?
It seems as a semi-lattice-like or upset-like structure. If I am not mistaken any $K$ has to be a union of subsets of $2^A$, which are themselves closed under union, while their infinitary union is $\emptyset$.
But this seems like a mouthful for something so intuitive. Maybe there is a name for this object or property, or some transformation to some property that has a name?