Wednesday, December 31, 2014

Structural specialization

There are a variety of cases in which it makes sense to preorder a set based upon the elements that it contains. One example is that with the set representation of a multiset the elements of the set include special multiple elements which are dependent upon previous multiples. Also algebraic structures like graphs have a specialization preorder associated with them in which edge elements are dependent upon the vertices that they contain. Multigraphs are a combination of these two notions as they can have multiples of edges which are dependent upon previous multiples of edges which are then dependent upon vertices.

A standard specialization relation can be provided that combines all these different notions of structures on sets into a single unified relation. This combined specialization relation will allow for all elements of distributive lattices to be treated like sets, multisets, and algebraic structures to be treated in a uniform way. This will then improve the handling of multisets and related structures like multigraphs so that they can be treated as a first class object in the algebra system.

Tuesday, December 16, 2014

Multiset systems

One of the critical problems of set theory is how to represent sequences as sets. Well Kuratowski solved the problem for sequences of size two through the set theoretic definition of the ordered pair there is no obvious solution to the problem of representing sequences of arbitrary size as set systems because sequences may contain repeated elements. I have thought about this problem so much in terms of set systems that I did not consider the possibility of using a multiset system instead.

In order to represent any sequence all we need to do is use a multiset system containing the multiset of elements up to a point in the sequence for each point in the sequence. This solution is so simple I am surprised I did not hear about this before. It is probably because multisets are far too often pushed aside for sets but no more. From now on I am going to make full use of multisets when I think about mathematics.