Tuesday, 14 November 2000
2:30 - 4:00    M Zawadowski
               Model completions and propositional logics

ABSTRACT:
In my talk I will report on my work with Silvio Ghilardi.

The orgin of our work goes back to the Theorem of A.M.Pitts, saying
that the second order intuitionistic calculus can be interpreted into the
ordinary intuitionistic propositional calculus.  It can be rephrased
in more categorical terms as follows: the dual of the category of finitely
presented Heyting algebras is a Heyting category.  Using this
formulation we have defined a 1'st order theory, in the laguage of Heyting
algebras which proves the same open formulas as the theory of Heyting algebras
$T_H$ and eliminates quantifies, i.e. we have shown that $T_H$ has a model
completion $T_H^*$.

We have shown that for many equational theories $T$ arisning from
logic the existence of the model completion $T^*$ is equivalent to the
fact that the dual of the category of finitely presented $T$-algebras
is a Heyting category. Thus the existence of the model completion
is equivalent to the existence of an additional categorical structure
on the dual of the category of fintely presented algebras.

Using a convinient description (duality) of the categories of finitely
presented Heyting and modal algebras we have identified those theories
of Heyting and modal algebras that admit model completion.  The
dualities allow us to prove and disprove some other properties of
those theories and corresponding logics.