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

Félix Bou, Àngel García-Cerdaña, Ventura Verdú
2005 Archive for Mathematical Logic  
The 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 FL ew by Ono), etc. In this paper we study the ∨, * , ¬, 0, 1 -fragment and the ∨, ∧, * , ¬, 0, 1 -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
more » ... iders the full consequence relation admitting hypotheses. It results that this notion of fragment is axiomatized by the rules of the sequent calculus FL ew 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.
doi:10.1007/s00153-005-0324-9 fatcat:a3sqyaylw5gmdmhdv5ocalzmtm