As reported in [32], a similar generalization has been previously used in 1993 by Guan and Bell [14] for the Dempster-Shafer rule using propositions in sequential logic and reintroduced in 1994 by Paris in his book [20],
page 4.

## Example Of A Super-Power Set:

Let's consider the frame Θ = {θ1, θ2} and let's assume θ1∩θ2 6= ∅, i.e. θ1 and θ2 are not disjoint according to Fig. 1 where A , p1 denotes the part of θ1 belonging only to θ1 (p stands here for *part*), B , p2 denotes the part of θ2 belonging only to θ2 and C , p12 denotes the part of θ1 and θ2 belonging to both. In this example, S
Θ={θ1,θ2}is then given by S
Θ = {∅, θ1 ∩ θ2, θ1, θ2, θ1 ∪ θ2, c(∅), c(θ1 ∩ θ2), c(θ1), c(θ2), c(θ1 ∪ θ2)}
where c(.) is the complement in Θ. Since c(∅) = θ1 ∪ θ2 and c(θ1 ∪ θ2) = ∅, the super-power set is actually given by S
Θ = {∅, θ1 ∩ θ2, θ1, θ2, θ1 ∪ θ2, c(θ1 ∩ θ2), c(θ1), c(θ2)}
Let's now consider the minimal refinement Θref = {**A, B, C**} of Θ built by splitting the granules θ1 and θ2 depicted on the previous Venn diagram into disjoint parts (i.e. Θref satisfies the Shafer's model) as follows:

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

Fig. 1: Venn diagram of a free DSm model for a 2D frame.

$$\cup\,C,\qquad\theta_{2}=B\cup C,$$

θ1 = A ∪ C, θ2 = B ∪ **C, θ**1 ∩ θ2 = C
7
Then the classical power set of Θref is given by

$$2^{\Theta^{r e f}}=\{\emptyset,A,B,C,A\cup B,A\cup C,B\cup C,A\cup B\cup C\}$$

We see that we can define easily a one-to-one correspondence, written ∼, between all the elements of the super-power set S
Θ and the elements of the power set 2 Θref as follows:

$\emptyset\sim\emptyset,\quad(\theta_{1}\cap\theta_{2})\sim C,\quad\theta_{1}\sim(A\cup C),\quad\theta_{2}\sim(B\cup C),\quad(\theta_{1}\cup\theta_{2})\sim(A\cup B\cup C)$
$$c(\theta_{1}\cap\theta_{2})\sim(A\cup B),\quad c(\theta_{1})\sim B,\quad c(\theta_{2})\sim A$$
Such one-to-one correspondence between the elements of S
Θ and 2 Θref can be defined for any cardinality |Θ| ≥ 2 of the frame Θ and thus one can consider S
Θ as the mathematical construction of the power set 2 Θref of the minimal refinement of the frame Θ. Of course, when Θ already satisfies Shafer's model, the hyper-power set and the super-power set coincide with the classical power set of Θ. It is worth to note that even if we have a mathematical tool to built the minimal refined frame satisfying Shafer's model, it doesn't mean necessary Θ