On the representation of (weak) nilpotent minimum algebras

We take a glimpse at the relation between WNM-algebras (algebraic models of the well-known Weak Nilpotent Minimum logic) and quasi-Nelson algebras, a non-involutive generalisation of Nelson algebras (models of Nelson's constructive logic with strong negation) that was introduced in a recent paper. We show that the two varieties can be related via the twist-structure construction, obtaining a new representation for a subvariety of WNM-algebras that includes the involutive ones (i.e. NM-algebras). Our results imply, in particular, that every pre-linear quasi-Nelson algebra is a WNM-algebra; we thus generalize the known result that the class of pre-linear Nelson algebras coincides with that of NM-algebras (models of Nilpotent Minimum logic).