Implicit Operations in MV-Algebras and the Connectives of Lukasiewicz Logic

It is shown that a conservative expansion of in…nite valued ×ukasiewicz logic by new connectives univocally determined by their axioms does not necessarily have a complete semantics in the real interval [0,1]. However, such extensions are always complete with respect to valuations in a family of MV-chains, Rational ×ukasiewicz logic being the largest one that has a complete semantics in [0,1]. In addition, the latter logic does not admit expansions by axiomatic implicit connectives that are not already explicit. Similar results are obtained for n-valued ×ukasiewicz logic and for the logic of abelian lattice ordered groups. These and related results are obtained by the study of compatible operations implicitly de…ned by identities in the varieties of MV-algebras and abelian `-groups; the pertaining algebraic results having independent interest.