Definition of cross-domain indexes and ordering functions in relational algebra and its usage in relational database management systems

In this thesis, a mathematical model that describes a “Unique Constraint Domain” is defined. Following, the “Ordered Unique Constraint Domain” is also mathematically defined. With those definitions, a cross-domain ordering is also defined. Then it is shown that relationships between tables in a Relational Database Management System can be defined in other forms than the usual ways, using cross-domain indexes, based in cross-domain ordering. It is shown that all foreign keys in a database can be transformed in indexes with the benefit of speeding data access. It is also shown that this technique is consistent with actual modelling techniques. It is shown how the index structure, with indexes defined as functions, can provide support for relationship roles. In addition, it is also shown how this can provide support for more than two tables in one relationship and for supporting special sorting order. The addition of a mathematical function to a relation that could sort that relation, demonstrating that the closure property of relations are still kept, shows that this mathematical model can be used as extension of the base relational model. Next, it is shown that with this new technique, commercial database engines should not degrade performance because all supporting structures are already present and, in some cases, a better performance might be achieved. Code for a prototype based in a Commercial Database Engine has been added, as an annexe, to show how this new technique can be used. Finally, future work can be done in this area considering that objects other than text and number need to be sorted (e.g. images, videos, audio data) developing new ways to find semantics to define order. The major contribute is the mathematical definition of the “Unique Constraint Domain” and the “Ordered Unique Constraint Domain” since they are mathematical models for candidate and primary keys.