

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

fmb(x
′). Hence, a monotone boolean function is defined by the values of the 2 belements (x1**, ..., x**b), and there is |Db| different monotone boolean functions.

All the values of all these monotone boolean function can be represented by a |Db| × 2 b matrix. If we multiply this matrix by the vector of all the possible intersections of the singletons in Θ with |Θ| = b (there is 2bintersections) given an union of intersections, we obtain all the elements of DΘ. We can also use the basis of all the unions of Θ (and obtain the intersections of unions), but with our codification the unions are coded with more integer numbers. So, the intersection basis is preferable.

Moreover, if we have some constraints (such as θ1 ∩ θ2 ≡ ∅), some elements of the intersection basis can be empty. So we only need to generate a |Db| × nb matrix where nb is the number of non empty intersections of elements in Θr.

For example, with the constraint given in example for |Θ| = 3, the basis is given by: ∅, θ1, θ2, θ3, θ1 ∩ θ3, θ2 ∩ θ3, and there is no θ1 ∩ θ2 and θ1 ∩ θ2 ∩ θ3.

Hence, the generation of DΘ
rcan run very fast if the basis is small, *i.e.* if there is some constraints. The Matlab code is given in section 5.

## 3.7 Decoding

Once the decision on one element A of DΘ
ris taken, we have to transmit this decision to the human operator. Hence we must to decode the element A (given by the integer numbers of the codification) in terms of unions and intersections of elements of Θ. If we know that A is in a subset of elements of DΘ
r given by the operator, we only have to scan this subset. Now, if the decision A comes from the focal elements (*a priori* unknown) or from all the elements of DΘ
r we must scan all DΘ
r with possibly high complexity. What we propose here is to consider the elements of DΘ
r ordering with first the elements most encountered in applications. Hence, we first scan the elements of 2Θ and in the same time the intersection basis that we must build for the generation of DΘ
r. Then, only if the element is not found we generate DΘ
r and stop the generation when found
(see the section 5 for more details).

The Smarandache's codification is an alternative to the decoding because