 a) Division as an internal operator: / : $L\times L\to L.$ Let $j\neq0,$ then ? 
$$L_{i}/L_{j}=\begin{cases}L_{[(i/j)\cdot(n+1)]}&\text{if}[(i/j)\cdot(n+1)]<n+1,\\ L_{n+1}&\text{otherwise.}\end{cases}$$
$$[(i/j)\cdot(n+1)]<n+1,$$
$$L_{i}/L_{j}={\frac{i/(n+1)}{j/(n+1)}}={\frac{(i/j)\cdot(n+1)}{n+1}}=L_{[(i/j)\cdot(n+1)]}$$
$$L_{i}-L_{j}=\begin{cases}L_{i-j}&\text{if}\quad i\geq j,\\ -L_{j-i}&\text{if}\quad i<j.\end{cases}\tag{11}$$  where $-L=\{-L_{1},-L_{2},\cdots,-L_{n},-L_{n+1}\}$. The $q$-subtraction above is well justified since when $i\geq j$, one has $L_{i}-L_{j}=\frac{i}{n+1}-\frac{1}{n+1}=\frac{1}{n+1}$.  
$$(12)$$
LiLjLk Lp + Lq = L(ijk)/(n+1)2 Lp+q= L(ijk)/(n+1)2 p+q·(n+1) = L(ijk)/(n+1) p+q= L ijk (n+1)(p+q)(12)
the integer part of the index.  
The first equality in (9) is well justified because when [(i/j) · (n + 1)] < n **+ 1, one has**
For example, if we have L0, L1, L2, L3, L4, L5, corresponding to respectively 0, 0.2, 0.4, 0.**6, 0**.8, 1, then: L1/L3 = L[(1/3)·5] = L[5/3] = L[1.66] ≈ L2. L4/L2 = L[(4/2)·5] = L[2·5] = Lmax = L5 **since**
10 > 5.

b) Division as an external operator: ⊘ : L×L → R
+. Let j 6= 0. Since Li⊘Lj = (i/(n+1))/(j/(n**+1)) =**
i/j**, we simply define**
Li ⊘ Lj = i/j **(10)**
Justification of b): when we divide say L4/L1 in the above example, we get 0.8/0.**2 = 4, but no**
label is corresponding to number 4 which is not even in the interval [0, **1], hence in the division as**
an internal operator we need to get as response a label, so in our example we approximate it to Lmax = L5**, which is a very rough approximation! So, depending on the fusion combination rules, it**
might better to consider the qualitative division as an external operator, which gives us the exact result.

- q-subtraction of linguistic labels: − : L × L → {L, −L}, The above qualitative operators are logical, justified due to the isomorphism between the set of linguistic equidistant labels and a set of equidistant numbers in the interval [0, **1]. These qualitative operators are built** exactly on the track of their corresponding numerical operators, so they are more mathematical than the ad-hoc definition of qualitative operators proposed so far in the literature. They are similar to the PCR5 combination numerical rule with respect to other fusion combination numerical rules based on the conjunctive rule. But moving to the enriched label qualitative operators the accuracy decreases.

Remark about doing multi-operations on labels**: When working with labels, no matter how many operations we have, the best (most accurate) result is obtained if we do only one approximation, and that one should**
be just at the very end. For example, if we have to compute terms like LiLjLk/(Lp + Lq**) as for qPCR5 (see**
example in section 6), we compute all operations as defined above, but without any approximations (i.e. not even calculating the integer part of indexes, neither replacing by n **+ 1 if the intermediate results is bigger than** n **+ 1), so:**
and now, when all work is done, we compute the integer part of the index, i.e. [ ijk
(n**+1)(**p+q)
] or replace it by n+ 1 if the final result is bigger than n + 1. Therefore, the term LiLjLk/(Lp + Lq**) will take the linguistic value** Ln+1 whenever [ ijk
(n**+1)(**p+q)
] > n **+ 1. This method also insures us of a unique result, and it is mathematically closer**
to the result that would be obtained if working with corresponding numerical masses. Otherwise, if one does approximations either at the beginning or after each operation or in **the middle of calculations, the inaccuracy** propagates (becomes bigger and bigger) and we obtain different results, depending on the places where the approximations were done.

$$({\mathfrak{g}})$$

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