Abstract
We give a representation theorem for Hilbert algebras by means of ordered sets and characterize 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 theorem. 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.
Highlights
Introduction and preliminariesHilbert algebras correspond to the algebraic counterpart of the implicative fragment of Intuitionistic Propositional Logic
We prove a characterization of the semi-homomorphism by means of deductive systems and we prove a characterization of the homomorphism by means of irreducible deductive systems
We give a representation theorem for Hilbert algebras and we give several results relating homomorphism of Hilbert algebras to deductive systems and semi-homomorphisms
Summary
We give a representation theorem for Hilbert algebras by means of ordered sets and characterize 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 theorem. Our central aim is to prove a new representation theorem for Hilbert algebras by means of ordered sets and give some results relating homomorphisms of Hilbert algebras to deductive systems and a weaker version of homomorphisms. See [2] or [4]
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