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 WNM, the variety of WNM-algebras (double-struck Wdouble-struck Ndouble-struck M) and 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 we study their standard completeness properties. We also characterize the generic WNM-chains, i. e. those that generate the variety double-struck Wdouble-struck Ndouble-struck M, and we give finite axiomatizations for some t-norm based extensions of WNM.