Mathematical Logic Quarterly 58 (3):188-207 (2012)

Ramon Jansana Ferrer
Universitat de Barcelona
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
Keywords 03G25  06A12  06F35  MSC (2010) 03G10  implicative semilattices  free extensions  Hilbert algebras
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DOI 10.1002/malq.201020098
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Implicative Semi-Lattices.William C. Nemitz - 1966 - Journal of Symbolic Logic 31 (2):273-273.

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