On triangular norm based axiomatic extensions of the weak nilpotent minimum logic

In this paper we carry out an algebraic investigation of the Weak Nilpotent Minimum logic (WNM) and its t-norm based axiomatic extensions. We consider the algebraic counterpart of this logic, the variety of WNM-algebras (WNM) and we prove that it is locally finite, so all its subvarieties are generated by finite chains. We give criteria to compare varieties generated by finite families of WNM-chains, in particular varieties generated by standard WNM-chains, or equivalently t-norm based axiomatic extensions of WNM, and study their standard completeness properties. We also characterize the generic WNMchains, i.e. those that generate the variety WNM, and we give finite axiomatizations for some t-norm based extensions of WNM.