Abstract
Let be a bounded Hilbert algebra and a -closed subset of . The Hilbert algebra of fractions is studied regarding maximal and irreducible deductive systems. As important results, we can mention a necessary and sufficient condition for a Hilbert algebra of fractions to be local and the characterization of this kind of algebras as inductive limits of some particular directed systems.
Highlights
The positive implication algebras characterize positive implicative logic, as it is mentioned in 1
The notion of deductive system, defined by Monteiro, is equivalent to the notion of implicative filter used by Rasiowa in 1
The maximal and irreducible deductive systems play an important role in the study of these algebras as the representation theorem for Hilbert algebras see Theorem 4.4 states
Summary
The positive implication algebras characterize positive implicative logic, as it is mentioned in 1 The theory of this kind of algebras has been developed by Diego in 2. Maximal deductive systems are studied by Busneag in 3 He is the one who studied the Hilbert algebras of fractions with respect to a -closed subset in 4, 5. There are established some properties referring to maximal and irreducible deductive systems These particular deductive systems are connected to -closed subsets. In the last section we study the Hilbert algebras of fractions, specially the one corresponding to maximal deductive systems. 6 , we define a presheaf on the base of the topological space Max A and we prove that a local Hilbert algebra of fractions is isomorphic to an inductive limit of a directed system.
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