Finitary Extensions of the Nilpotent Minimum Logic and (Almost) Structural Completeness

Abstract

In this paper we study finitary extensions of the nilpotent minimum logic (NML) or equivalently quasivarieties of NM-algebras. We first study structural completeness of NML, we prove that NML is hereditarily almost structurally complete and moreover NM $$^{-}$$ , the axiomatic extension of NML given by the axiom $$\lnot (\lnot \varphi ^{2})^{2}\leftrightarrow (\lnot (\lnot \varphi )^{2})^{2}$$ , is hereditarily structurally complete. We use those results to obtain the full description of the lattice of all quasivarieties of NM-algebras which allow us to characterize and axiomatize all finitary extensions of NML.