## Pivot and Unpivot

### October 14, 2010

SQL has many flaws, and the fact that its syntax poorly reflects duality of pivot and unpivot queries is probably somewhere at the bottom of the list. To be fair, relational algebra doesn’t do any better. What are pivot and unpivot queries in QBQL?

Consider the following matrix example:

$\begin{bmatrix} 3 & 1 \\1 & 0 \end{bmatrix}$

This 2×2 matrix can be represented relationally in two ways:
• as ternary relation with two subscripts and one entry attribute:

Au=[i j s]
1 1 3
1 2 1
2 1 1
2 2 0
;


• as rectangular array with dimension j pivoted into the columns j1 and j2:

Ap=[i j1 j2]
1 3  1
2 1  0
;


The expressing Ap in terms of Au is pivot operation, while representing Au in terms of Ap is unpivot.

Au =
((Ap /^ [j1=s]) /^ [j]1)
v
((Ap /^ [j2=s]) /^ [j]2)
.

Ap =
((Au /^ [j1=s]) /^ [j]1)
^
((Au /^ [j2=s]) /^ [j]2)
.


Notice that both expressinos are almost identical, so the duality of pivot and unpivot is exposed as formal relational lattice duality of join and union.