Principal moore families

Principal ideal domains, principal ideal rings, etc are set system theoretic classifications of systems of ideals of certain rings or integral domains. This leads naturally to a set theoretic generalization: the concept of principally generated moore families of sets. These can be defined like so:

(defn principal-moore-family?
  [family]

  (and
   (moore-family? family)
   (let [singleton-closures (set
                             (map
                              (fn [i]
                                (cl family #{i}))
                              (apply union family)))]
     (= family singleton-closures))))

As a general concept applicable to any kind of set system (or hypergraph) which emerges from algebra, topology, analysis, or any other subject these can also be applied to other constructs besides commutative rings. Sub(G) for a finite cyclic group also forms a principal moore family. Here are some examples of this predicate in action:

; true example
(principal-moore-family? 
  #{#{0} #{0 1} #{0 1 2 3}}) 

; false example
(principal-moore-family? 
  #{#{0} #{0 1} #{0 2} #{0 3} #{0 1 2 3}}) 

Some of these principal moore families also form preorder containment families, and they can potentially have interesting intersections with other classes of set systems which is a subject for further exploration.