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On the free implicative semilattice extension of a Hilbert algebra
Author(s) -
Celani Sergio A.,
Jansana Ramon
Publication year - 2012
Publication title -
mathematical logic quarterly
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.473
H-Index - 28
eISSN - 1521-3870
pISSN - 0942-5616
DOI - 10.1002/malq.201020098
Subject(s) - semilattice , mathematics , hilbert's basis theorem , extension (predicate logic) , algebra over a field , hilbert's fourteenth problem , ideal (ethics) , distributive property , algebraic semantics , pure mathematics , discrete mathematics , algebraic number , rigged hilbert space , hilbert space , computer science , mathematical analysis , philosophy , semigroup , epistemology , programming language , unitary operator
Abstract Hilbert algebras provide the equivalent algebraic semantics in the sense of Blok and Pigozzi to the implication fragment of intuitionistic logic. They are closely related to implicative semilattices. Porta proved that every Hilbert algebra has a free implicative semilattice extension. In this paper we introduce the notion of an optimal deductive filter of a Hilbert algebra and use it to provide a different proof of the existence of the free implicative semilattice extension of a Hilbert algebra as well as a simplified characterization of it. The optimal deductive filters turn out to be the traces in the Hilbert algebra of the prime filters of the distributive lattice free extension of the free implicative semilattice extension of the Hilbert algebra. To define the concept of optimal deductive filter we need to introduce the concept of a strong Frink ideal for Hilbert algebras which generalizes the concept of a Frink ideal for posets.