  Motivated by Kalman residuated lattices, Nelson residuated lattices and
Nelson paraconsistent residuated lattices, we provide a natural common
generalization of them. Nelson conucleus algebras unify these examples and
further extend them to the non-commutative setting. We study their structure,
establish a representation theorem for them in terms of twist structures and
conuclei that results in a categorical adjunction, and explore situations where
the representation is actually an isomorphism. In the latter case, the
adjunction is elevated to a categorical equivalence. By applying this
representation to the original motivating special cases we bring to the surface
their underlying similarities.
