dear Aldo
it is clear that <<I>> is not the only solution ; you rightly propose  
a generalization, which by the way implies that you don't doubt  
soundness of my interpretation. One should surely do this to  
accomodate several exponentials. But my choice is already complete :  
simply take the usual monoid of contexts -but ignore the multiplicity  
of formulas ?A- ; it is plain that idempotents are exactly the  
contexts ?\Gamma and that they are all in the fact <<1>>. From this  
completeness follows.
So there is no mistake in my definition, only the apparent conflict  
- the essential non-unicity of <<!>>
- the unicity of my definition of <<I>>
But one should remember that the semantics enables one to make  
constructions that have no direct meaning in terms of syntax. 

I am trying to give a possible explanation : let us define the  
congruence p~q by {p}^\perp = {q}^\perp. Let define <<J>> as the set  
of elements of <<1>> which are such that p~p^2. <<J>> works too, but  
without completeness ; should I have used <<J>>, I would now be in  
the pit you expected, since <<J>> has a kind of logical meaning. The  
actual completeness could be stated in terms of arbitrary subsets of  
<<J>>, closed under product, and one can restrict to free monoids.  
However, the model is not significantly altered if one quotients it  
by any congruence coarser than <<~>>. Now, if <<K>> is an arbitrary  
subset of <<J>>, introduce the congruence generated by the  
equivalence between p and p^2 for p \in K. Then we discover that -in  
the quotient monoid- equality p=p^2 is just another way to speak of  
<<K>>, i.e. what is arbitrary in the choice of <<K>> can be hidden in  
the equality of the monoid. 


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