A n-ary relation is a set of sequences that each have n slots with values associated with them. Functional dependencies can be used to describe how different slots in relation are dependent upon one another, so for example in a binary relation we can describe that the second value is dependent upon the first, which is the typical definition of a unary operation. This produces a closure operation on the set of slots, so that given some set of slots, the closure can be used to produce the information which can be inferred from them.
Closure operations are a common part of mathematics, and they are associated with lattices and moore families. For example, given a group the subgroup generated by a set is a closure operation, and in particular in linear algebra the span generated by a set of vectors is a closure operation. Also in linear algebra, the dimension is the smallest number of generators needed to span the entire vector space. It is for this reason that I call the functional dimension the number of slots needed to generate all the elements of a relation.
The interesting thing is that this is a monotone function function like cardinality, so the subrelations of a given n-ary relation must have less functional dimension then their parents. This monotone function can be used to help to classify relations and their ordering.
No comments:
Post a Comment