The Localization of Commutative (unbounded) Hilbert Algebras

Hilbert algebras are important tools for certain investigations in algebraic logic since they can be considered as fragments of any propositional logic containing a logical connective implication and the constant 1 which is considered as the logical value “true”. The concept of Hilbert algebras was introduced in the 50-ties by L. Henkin and T. Skolem (under the name implicative models) for investigations in intuitionistic logics and other non-classical logics. Diego [9] proved that Hilbert algebras form a variety which is locally finite. They were studied from various points of view. In this paper we develop a theory of localization for commutative (unbounded) Hilbert algebras, and then we deal with generalizations of results which are obtained in the papers [4] and [5] for bounded case. This paper is organized as follows: In Section 2 we recall the basic definitions and put in evidence many rules of calculus in commutative Hilbert algebras which we need in the rest of paper (especially c11–c14). In Section 3 we introduce the commutative Hilbert algebra of fractions relative to a ∨-closed system. In Section 4 we develop a theory for multipliers on a commutative (unbounded) Hilbert algebra. In Section 5 we define the notions of Hilbert algebras of fractions and maximal Hilbert algebra of quotients for a commutative (unbounded) Hilbert algebra. In the least part of this section is proved the existence of the maximal Hilbert algebra of quotients (Theorem 21). In Section 6 we develop a theory of localization for commutative (unbounded) Hilbert algebras. So, for a commutative (unbounded) Hilbert algebra A we