Storage Operators and Multiplicative Quantifiers in Many-valued Logics

To every many-valued logic L we associate a logic LS obtained from L by the adding of a storage operator(*) which has some analogies with Girard's exponential !. The algebraic counterpart of * is the operator I associating to any element a of a many-valued algebra the greatest idempotent below a. We investigate the algebraic completeness of the logics obtained in this way, as well as completeness with respect to real and rational-valued semantics. For some of these logics, we also discuss decidability and complexity. In the predicate case, the storage operator allows one to obtain a multiplicative universal quantifier which can be regarded as the iteration of the multiplicative conjunction, in the same way as the usual universal quantifier can be regarded as the iteration of the additive conjunction.