$$q S_{3}(X)\triangleq\sum_{\begin{array}{l}{{X_{1},X_{2},\ldots,X_{k}\in D^{\Theta}}}\\ {{X_{1}\cup X_{2}\cup\ldots\cup X_{s}=X}}\\ {{X_{1}\cap X_{2}\cap\ldots\cap X_{s}\in\Theta}}\end{array}}\prod_{i=1}^{s}q m_{i}(X_{i})$$

$$(32)$$
qmi(Xi) (32)
with U , u(X1)∪**. . .**∪u(Xs) where u(X) is the union of all θithat compose X, It , θ1 ∪**. . .**∪θn is the total ignorance. qS1(X) is nothing but the qDSmC rule for s independent sources based on Mf(Θ); qS2(X) is the qualitative mass of all relatively and absolutely empty sets which is transferred to the total or relative ignorances associated with non existential constraints (if any, like in some dynamic problems); qS3(X) transfers the sum of relatively empty sets directly onto the canonical disjunctive form of non-empty sets. qDSmH generalizes qDSmC works for any models (free DSm model, Shafer's model or any hybrid models) when manipulating qualitative belief assignments.

## 6.6 Qualitative Pcr5 Rule (Qpcr5)

In classical (i.e. quantitative) DSmT framework, the Proportional Conflict Redistribution rule no. 5 (PCR5)
defined in [34] has been proven to provide very good and coherent results for combining (quantitative) belief masses, see [9,33]. When dealing with qualitative beliefs within the DSm Field and Linear Algebra of Refined Labels [36] we get an exact qualitative result no matter what fusion rule is used (DSm fusion rules, Dempster's rule, Smets's rule, Dubois-Prade's rule, etc.). The exact qualitative result will a refined label (but the user can round it up or down to the closest integer index label).

## 6.7 A Simple Example Of Qualitative Fusion Of Qba'S

Let's consider the following set of ordered linguistic labels

$$L=\{L_{0},L_{1},L_{2},L_{3},L_{4},L_{5}\}$$

(for example, L1, L2, L3 and L4 may represent the values: L1 , very poor, L2 , poor, L3 , *good* and L4 , *very good*, where , symbol means *by definition*).

Let's consider now a simple two-source case with a 2D frame Θ = {θ1, θ2}, Shafer's model for Θ, and qba's expressed as follows:

$$q m_{1}(\theta_{1})=L_{1},\quad q m_{1}(\theta_{2})=L_{3},\quad q m_{1}(\theta_{1}\cup\theta_{2})=L_{1}$$
$$q m_{2}(\theta_{1})=L_{2},\quad q m_{2}(\theta_{2})=L_{1},\quad q m_{2}(\theta_{1}\cup\theta_{2})=L_{2}$$

The two qualitative masses qm1(.) and qm2(.) are normalized since:
qm1(θ1) + qm1(θ2) + qm1(θ1 ∪ θ2) = L1 + L3 + L1 = L**1+3+1** = L5 and qm2(θ1) + qm2(θ2) + qm2(θ1 ∪ θ2) = L2 + L1 + L2 = L**2+1+2** = L5 We first derive the result of the conjunctive consensus. This yields:

$$qm_{12}(\theta_{1})=qm_{1}(\theta_{1})qm_{2}(\theta_{1})+qm_{1}(\theta_{1})qm_{2}(\theta_{1}\cup\theta_{2})+qm_{1}(\theta_{1}\cup\theta_{2})qm_{2}(\theta_{1})$$ $$=L_{1}\times L_{2}+L_{1}\times L_{2}+L_{1}\times L_{2}$$ $$=L_{\frac{1,2}{5}}+L_{\frac{1,2}{5}}+L_{\frac{1,2}{5}}=L_{\frac{2}{5}+\frac{2}{5}+\frac{2}{5}}=L_{\frac{6}{5}}=L_{1,2}$$
$$qm_{12}(\theta_{2})=qm_{1}(\theta_{2})qm_{2}(\theta_{2})+qm_{1}(\theta_{2})qm_{2}(\theta_{1}\cup\theta_{2})+qm_{1}(\theta_{1}\cup\theta_{2})qm_{2}(\theta_{2})$$ $$=L_{3}\times L_{1}+L_{3}\times L_{2}+L_{1}\times L_{1}$$ $$=L_{\frac{\theta_{1}}{5}}+L_{\frac{\theta_{2}}{5}}+L_{\frac{1+1}{5}}=L_{\frac{\theta_{2}}{5}+\frac{\theta_{1}}{5}+\frac{1}{5}}=L_{\frac{10}{5}}=L_{2}$$
