Regular families

Regular families are families for which every member of the union of the sets of the family has the same degree. They are a special case of the classification of set systems by their degree multisets (the ontology of multisets determines the classification by degrees). So a regular multiset by comparison is a multiset in which the multiplicity of each element is equal for example {a,a,a,b,b,b,c,c,c} is regular because the multiplicity of each element is equal. The ontology of regular families is included below.



Special cases include the rank complete families, which are set systems determined by the collection of all subsets of some set, typically produced with the selections function and the union of such sets. Another special case is the case of independent families (families whose degree multisets are actually sets which works because sets are regular multisets) which is a different classification by degrees. Similar concepts of regular families emerge from the set theoretic representation of the graphs (either maximal cliques families or dependency families).