Involutive Commutative Residuated Lattice without Unit: Logics and Decidability

We investigate involutive commutative residuated lattices without unit, which are commutative residuated lattice-ordered semigroups enriched with a unary involutive negation operator. The logic of this structure is discussed and the Genzten-style sequent calculus of it is presented. Moreover, we prove the decidability of this logic.