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We want here to embed fuzzy set theory in a larger framework called the mnesor theory and based on the semimodule over the positive real numbers.

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## Ii. Semimodule Over The Positive Real Numbers

R+) is considered with two operations: the The set of the positive real numbers (denoted multiplication and the maximum operator (written 
⊕) which will be called addition: for any x, y ∈ R+ , 
x ⊕ y = max(x, y).

s 0 . We add an external multiplication returning Let M denote a commutative monoid with the identity element 
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λ ∈ R+. 

R+
Aλ ∈ M from A ∈ M and M is called a semimodule over 
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λ,µ ∈ R+ and iff the four next properties hold for any A,B ∈ M :

$$(1)$$
$$A\,1=A$$
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$$\begin{array}{c}{{(2)}}\\ {{(3)}}\\ {{(4)}}\end{array}$$
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$$\begin{array}{r l}{A\,\lambda+A\,\mu=A\left(\lambda\oplus\mu\right)}\\ {\left(A+B\right)\lambda=A\,\lambda+B\,\lambda}\\ {\left(A\,\lambda\right)\mu=A\left(\lambda\times\mu\right)}\end{array}$$

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We first derive some basic properties.

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Idempotence. The addition of fuzzy mnesors is idempotent. PROOF. For any A ∈ M , 
A + A = A1+ A1= A(1 ⊕1) = A1 = A
Ordering. An order relation is naturally defined by the addition: 
A ⊆ B iff A + B = B.

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PROOF. 

⊆ is an order relation indeed, since M is a commutative idempotent monoid.

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Note that if B = Aλ with λ ≤1, then B is inferior to A (
A + Aλ = A1+ Aλ = A(1 ⊕ λ) = A1 = A
) and if B = Aλ with λ ≥1, then B is superior to A. In the fomer case, we say that B is more 
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