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$$\begin{array}{l}{{\vDash x\left(0\right)=x}}\\ {{\vDash\left(x+y\right)\lambda=x\,\lambda+y\,\lambda}}\\ {{\vDash\left(x\,\lambda\right)\mu=x\left(\lambda\otimes\mu\right)}}\\ {{\vDash x\left(\lambda\oplus\mu\right)=x\,\lambda+x\,\mu}}\end{array}$$
x (0) = x (1)
(x + y)λ = x λ + y λ (2)
(x λ)µ = x (λ ⊗µ) (3)
x (λ ⊕µ) = x λ + xµ (4)
$$\begin{array}{l}{(5)}\\ {(6)}\end{array}$$
x = x (5)
$${\frac{\overline{{{\chi}}}}{x\,\lambda}}=\overline{{{\chi}}}\,\chi^{-1}$$
$\square$
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(x × y) λ = (x λ) ×(y λ),  € € x × (x λ) = x λ, with  λ p (0)
PROOF. $\left(x\times y\right)\lambda=\overline{\left(\overline{x}+\overline{y}\right)}\lambda=\overline{\left(\overline{x}+\overline{y}\right)}\lambda^{-1}=\overline{x}\overline{\lambda}^{+}+\overline{y}\overline{\lambda}^{-1}=\overline{x}\overline{\lambda}+\overline{y}\overline{\lambda}=\left(x\lambda\right)\times\left(y\lambda\right)$.  
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That's formally what defines a semimodule over the flat numbers. The elements are named mnesors and precisely flat mnesors. Furthermore, we add a unary function (
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x → x the conjugate of 

$$\begin{array}{l}{{\mathrm{(1)}}}\\ {{\mathrm{(2)}}}\\ {{\mathrm{(3)}}}\end{array}$$

$\eqref{eq:walpha}$. 
x ) which is:
involutive, compatible with the external multiplication, 
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Idempotence. We prove first that the addition of mnesors is idempotent
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PROOF. In x + x , we substitute x τ for x . Thus x + x = x τ + x τ = x (τ ⊕τ ) = x τ = x Ordering. The ordering is defined as follows: the mnesor x is inferior to the mnesor y (written x ≤ y ) if and only if x can be written as x = y λ where λ p (0).

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PROOF. Obviously reflexive and transitive.

Antisymmetric: 
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x ≤ y and y ≤ x , that is x = y λ and y = xµ. Then 
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x + y = x + xµ = x (0) + xµ = x ((0) ⊕µ) = x (0) = x and x + y = y λ + y = y . Then x = y .

We prove next that for all x in Go, x λ ≤ xµ whenever λ p µ.

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Go returns one of the operands, for example we can write PROOF. As the addition in λ ⊕ µ = µ. 

So, x λ + xµ = x (λ ⊕µ) = xµ
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Empty mnesor. The mnesor identity 0 is called the empty mnesor and it verifies the identity 
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0 λ = 0 whenever λ p (0)
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PROOF. 

0 λ = 0 + 0 λ = 0(0) + 0λ = 0((0) ⊕ λ) = 0(0) = 0
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(Internal) multiplication. The product of the mnesors x and y is the mnesor x × y defined as x × y = x + y . The operation is associative, commutative and idempotent. Two properties hold:
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