Saturday, September 8, 2018

Theory of relations

The concept of a total ordering relation plays a fundamental role in mathematics. Sequences are a type of total ordering in which elements can occur more then once. Sequences in which no element appears more then once are no different then a total order. The importance of this concept derives from the importance of total orders. By combining the theory of total orders and sequences with set theory we can get the theory of relations which deals with sets of sequences.

Given a partial ordering relation, sequences in the partial order are called monotone non-decreasing if no two elements coming after one another are ever less then one another in the partial order. It is monotone increasing if they are not equal either. A partial ordering relation can be determined by its relation of monotone sequences. The typical definition of a partial ordering relation is the set of all monotone non-decreasing sequences in the partial order of size two. This relation of monotone non-decreasing sequences has the properties of reflexivity, anti-symmetry, and transitivity that characterize partial orders. This leads to a special kind of relation.

Given a partial order we can form a relation from its monotone non-decreasing sequences of a certain size. If it is finite height, we can form a relation consisting of its maximal monotone increasing sequences, and so on. In general, it is useful to consider relations consisting of sequences of any size using this understanding of order theory and sequences. The notion of a nary relation is merely a special case which consists of sequences of any particular size.

As a consequence, it is now useful to extend our understanding of relations beyond the theory of nary relations and specifically unary relations, binary relations, ternary relations, and quaternary relations, which is typically established. We can use concepts analogous to the theory of set systems to talk about special classes of relations. The rank of a relation is the size of its largest sequence. Sequences in a relation can be distinct or equal and so on. The theory of relations will play a foundational role in the construction of our new computer algebra system.

Monday, September 3, 2018

Total order theory

My recent research into the theory of order topology led me to realize just how connected the theory of order topology and metric spaces is. A metric space can be partially characterized by the order topology of its distances, so that a metric space with discrete distances is itself discrete. In this way, order topology is connected not only to the theory of metric spaces but all of mathematical analysis. The results of this research, and a considerable examination of the theory of total orders in general is presented in the following paper.

https://drive.google.com/file/d/1RNg0yXtIad6ziTvjkzINajBknPl6n9wG/view?usp=sharing