# Re: EXPONENTIAL



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
between
- 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.

Ciao
jy
---

Jean-Yves GIRARD
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