Antiregular families

Set systems can be classified by their degree multisets. Regular families are the set systems in which the degree of each union member is equal, antiregular families by contrast are the set systems for which the degree of every union member is different. Progression families like {{0} {0 1} {0 1 2} {0 1 2 3}} are antiregular because the degree of each union member is different from the others. Kuratowski pairs, rather they are equal like {{0}} or distinct like {{0} {0 1}} are also antiregular.



Antiregular families characterize the total orderings on a set (all preorders and graphs can be described as set systems), but not the total preorderings because preorders can have elements that are equal and indistinguishable. The total ordering on the set can be recovered from the multiset alone by the ordering of the multiplicities of the multiset, which is the basis of the multiset theoretic definition of the ordered pair. The corresponding progression family then be recovered from the multiset by the total ordering, which means antiregular families a special case of family that can be recovered directly from their multisets.

The same properties don't occur for sets of multisets, as sets of multisets can take a variety of forms besides progressions for example like {{0}, {1,1}}. Progressions of multisets which can be used to characterize sequences in pure multiset theory aren't always antiregular like with {{0},{0,1},{0,1,1}} which is equal to (0 1 1). This is actually a regular progression of multisets rather then an antiregular one which shows that this property of recoverability from the multiset does not apply sets of multisets like it does to sets of sets.