(rrrl; rrrl); llrl 6= rrrl; (rrrl; llrl).

For DRAfp all 512000 triples of base-relations are associative w.r.t. composition.

With this result, we obtain that DRAfp is a relation algebra in a strict sense.

## 3.8 Draf Composition Is Weak

The failure of DRAf to be associative may imply that its composition is also weak. We will investigate this in this section. First, recall the definition of strong composition. Furthermore, the composition of *OPRA*1 is known to be weak [49], but by Ex. 19 and Prop. 20, then DRAf also has a weak composition.

Definition 49. A Qualitative Composition is called *strong* if, for any arbitrary pair of objects A, C in the domain in relation AracC, there is for every entry in the composition table that contains AracC on the right hand side, an object B such that ArabB and BrbcC reconstruct this entry.

We will show now that the defining property of strong composition (see Sect. 2.3) is violated for DRAf . Proposition 50. The composition of DRAf *is weak.*
Proof. Consider the DRAf composition A BFII B; B LLLB C 7→
A LLLL C. We show that there are dipoles A and B such that there is no dipole B which reflects the composition. Consider dipoles A and B as shown in Fig. 25. We observe that they are in the DRAfp relation LLLL- with the dipole

![0_image_0.png](0_image_0.png)

C pointing towards the line dipole A lies on. Because of A BFII B, dipole B has to lie on the same line as A. But, since C is a straight line and lines A and B lie in front of C, the endpoint of B cannot lie behind C.

As expected, the composition of DRAf turns out to be weak. Let us have a closer look at the composition of DRAfp in the next section.

20In the master thesis of one of our students, a detailed analysis of a specific non-associative dipole configuration is presented [57]