What would one call a family $\mathcal{F}$ of subsets $X$ such that:
$$(1):X\in\mathcal{F}$$ $$(2):A,B\in\mathcal{F}\implies A\setminus B\in\mathcal{F}$$ $$(3):P\subseteq\mathcal{F}\text{ is a partition of }X\implies \forall Q\subseteq P(\cup_{S\in Q}S\in\mathcal{F})$$
Properties one and two are equivalently to saying the pair $(X,\mathcal{F})$ is a field of sets however the third property is distinct. Do these have a name? They are coming up when I try to extend the notion of a "connected space" to interior algebras.
Note $\mathcal{L}=(\mathcal{F},\subseteq,\cup,\cap)$ is a distributive lattice since finite unions/intersections are closed in $\mathcal{F}$ in fact the join irreducibles of $\mathcal{L}$ are the atoms of $\mathcal{L}$, thus $\mathcal{L}$ is atomistic iff every element of $X$ belongs to an inclusion minimal set in $\mathcal{F}\setminus\{\emptyset\}$ iff the sets in $\mathcal{F}$ are closed under arbitrary unions iff $(X,\mathcal{F})$ is a partition topology. What about if $(X,\mathcal{F})$ is not a partition topology?
Can these be characterized as the clopen sets of a particular class of topologies? If so, which ones?