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On two fragments with negation and without implication of the logic of residuated lattices

AutorBou, Felix; García-Cerdaña, Àngel ; Verdu, Ventura
Palabras clavePseudocomplemented monoids
Algebraizable logics
Gentzen systems
Residuated lattices
Substructural logics
Fecha de publicación2006
EditorSpringer
CitaciónArchive for Mathematical Logic 45: 615- 647 (2006)
ResumenThe logic of (commutative integral bounded) residuated lattices is known under different names in the literature: monoidal logic [26], intuitionistic logic without contraction [1], H BCK [36] (nowadays called [InlineMediaObject not available: see fulltext.] by Ono), etc. In this paper we study the [InlineMediaObject not available: see fulltext.]-fragment and the [InlineMediaObject not available: see fulltext.]-fragment of the logical systems associated with residuated lattices, both from the perspective of Gentzen systems and from that of deductive systems. We stress that our notion of fragment considers the full consequence relation admitting hypotheses. It results that this notion of fragment is axiomatized by the rules of the sequent calculus [InlineMediaObject not available: see fulltext.] for the connectives involved. We also prove that these deductive systems are non-protoalgebraic, while the Gentzen systems are algebraizable with equivalent algebraic semantics the varieties of pseudocomplemented (commutative integral bounded) semilatticed and latticed monoids, respectively. All the logical systems considered are decidable.
URIhttp://hdl.handle.net/10261/161162
Identificadoresdoi: 10.1007/s00153-005-0324-9
issn: 0933-5846
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