AbstractThe aim of this paper is to give, using the Kripke semantics for intuitionism, a representation of finitely generated free Heyting algebras. By means of the representation we determine in a constructive way some set of “special elements” of such algebras. Furthermore, we show that many algebraic properties which are satisfied by the free algebra on one generator are not satisfied by free algebras on more than one generator.
This note confirms that the existence proof for absolutely free algebras originated by Dedekind in [2] and completely developed for instance in [4] can still be carried out in a topos containing an infinite object i.e. an object N for which N ≃ N+1 if the type of the algebras considered is finite, pointed and internally projective i.e. is a finite sequence of objects, (Ij)i≤j≤k for which the functors ( )Ij preserve epimorphisms and each of which has a global section.