© Hindawi Publishing Corp. A NOTE ON HOMOMORPHISMS OF HILBERT ALGEBRAS

We give a representation theorem for Hilbert algebras by means of ordered sets and char-acterize the homomorphisms of Hilbert algebras in terms of applications defined between the sets of all irreducible deductive systems of the associated algebras. For this purpose we introduce the notion of order-ideal in a Hilbert algebra and we prove a separation theo-rem. We also define the concept of semi-homomorphism as a generalization of the similar notion of Boolean algebras and we study its relation with the homomorphism and with the deductive systems. 2000 Mathematics Subject Classification: 08A30. 1. Introduction and preliminaries. Hilbert algebras correspond to the algebraic counterpart of the implicative fragment of Intuitionistic Propositional Logic. For the general development of Hilbert algebras, the notion of deductive system (also called implicative filters) plays an important role. For example, it is known that the set of all deductive systems of a Hilbert algebra is a distributive lattice isomorphic to th