
## A. Power Set And Hyper-Power Set

In DST framework, one considers a frame of discernment Θ = {θ1**, . . . , θ**n} as a finite set of n exclusive and exhaustive elements (*i.e.* Shafer's model denoted M0(Θ)). The *power set* of Θ is the set of all subsets of Θ. The order of a power set of a set of order/cardinality |Θ| = n is 2 n. The power set of Θ is denoted 2 Θ. For example, if Θ = {θ1, θ2}, then 2 Θ = {∅, θ1, θ2, θ1 ∪ θ2}.

In DSmT framework, one considers Θ = {θ1**, . . . , θ**n} be a finite set of n exhaustive elements only (*i.e.* free DSm-model denoted Mf**(Θ))**. Eventually some integrity constraints can be introduced in this free model depending on the nature of the problem of interest. The hyper-power set of Θ (*i.e.* the free Dedekind's lattice) denoted DΘ [18] is defined as:
1) ∅, θ1**, . . . , θ**n ∈ DΘ.

2) If **A, B** ∈ DΘ, then A ∩ B, A ∪ B ∈ DΘ.
3) No other elements belong to DΘ, except those obtained by using rules 1 or 2.

If |Θ| = n, then |DΘ| ≤ 2 2 n. Since for any finite set Θ, |DΘ**| ≥ |**2 Θ|, we call DΘ the *hyper-power set* of Θ. For example, if Θ = {θ1, θ2}, then DΘ = {∅, θ1 ∩ θ2, θ1, θ2, θ1 ∪ θ2}. The *free DSm model* Mf(Θ) corresponding to DΘ allows to work with vague concepts which exhibit a continuous and relative intrinsic nature. Such concepts cannot be precisely refined in an absolute interpretation because of the unreachable universal truth.

It is clear that Shafer's model M0(Θ) which assumes that all elements of Θ are truly exclusive is a more constrained model than the free-DSm model Mf(Θ) and the power set 2 Θ can be obtained from hyper-power set DΘ by introducing in Mf(Θ)
all exclusivity constraints between elements of Θ. Between the free-DSm model Mf(Θ) and Shafer's model M0(Θ), there exists a wide class of fusion problems represented in term of the DSm hybrid models denoted M(Θ) where Θ involves both fuzzy continuous and discrete hypotheses. The main differences between DST and DSmT frameworks are (i) the model on which one works with, and (ii) the choice of the combination rule and conditioning rules [18], [19]. In the sequel, we use the generic notation GΘ for denoting either DΘ (when working in DSmT with free DSm model) or 2 Θ (when working in DST
with Shafer's model).

## B. Quantitative Basic Belief Assignment (Bba)

The (quantitative) basic belief assignment (bba) m(.) has been introduced for the first time in 1976 by Shafer [17] in his Mathematical Theory of Evidence (*i.e.* DST). m(.) is defined as a mapping function from 2 Θ → [0, 1] provided by a given source of evidence B satisfying the conditions:

2
$m(\emptyset)=0$,. 
$$(1)$$
m(∅**) = 0**, (1)
$$\sum_{A\in2^{\Theta}}m(A)=1.\tag{2}$$
The elements of 2 Θ having a strictly positive mass are called *focal elements* of B. The set of focal elements of m(.) is called the core of m(.) and is usually denoted F(m). The equation (1) corresponds to the closed-world assumption [17]. As