Self-Correlation and Maximum Independence in Finite Relations

We consider relations with no order on their attributes as in Database Theory. An independent partition of the set of attributes S of a finite relation R is any partition X of S such that the join of the projections of R over the elements of X yields R. Identifying independent partitions has many applications and corresponds conceptually to revealing orthogonality between sets of dimensions in multidimensional point spaces. A subset of S is termed self-correlated if there is a value of each of its attributes such that no tuple of R contains all those values. This paper uncovers a connection between independence and self-correlation, showing that the maximum independent partition is the least fixed point of a certain inflationary transformer α that operates on the finite lattice of partitions of S. α is defined via the minimal self-correlated subsets of S. We use some additional properties of α to show the said fixed point is still the limit of the standard approximation sequence, just as in Kleene’s well-known fixed point theorem for continuous functions.