Twist structures and Nelson conuclei

TL;DR

This paper introduces Nelson conucleus algebras, unifying various Nelson residuated lattice structures, and provides a representation theorem using twist structures and conuclei, establishing categorical relationships.

Contribution

It generalizes Nelson residuated lattices to a non-commutative setting and develops a representation theorem linking these algebras to twist structures and conuclei.

Findings

Established a categorical adjunction for Nelson conucleus algebras.

Proved conditions under which the adjunction becomes an isomorphism or equivalence.

Unified several Nelson residuated lattice variants through the new framework.

Abstract

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.