Algebraically closed and existentially complete p-algebras
by
Juerg Schmid
University of Bern, Switzerland

Abstracting from the classical notions of an algebraically closed resp. real closed field, Abraham Robinson developed, in the sixties, a general concept of an algebraically closed and existentially complete model (of a theory in a first-order language) and showed that such models exist under mild conditions on the theory.

In this talk, we review these concepts for so-called (distributive) p-algebras which are just the algebraic counterparts for intuitionistic propositional calculus, including Boolean algebras and thus classical propositional calculus as a special case. It was well-known to model theorists that different varieties of p-algebras (the so-called Lee classes) contain a uniquely determined countable existentially complete algebra; however, these algebras resisted an explicit description in most cases. We show how natural dualities may be used to obtain such a description in a nontrivial case. Finally, we consider briefly what happens if we drop the join operation in p-algebras and thus deal with so-called pseudocomplemented semilattices.

Date received: September 7, 2003