From partially ordered monoids to partially ordered groups via free nuclear preimages

TL;DR

This paper explores how nuclear and conuclear images can transform residuated lattices and pomonoids, establishing new links between these structures and partially ordered groups, with implications for non-classical logic.

Contribution

It introduces the concept of free nuclear preimages for pomonoids and characterizes certain classes of residuated lattices and pomonoids as nuclear images of group-based structures.

Findings

Finite commutative integral residuated lattices are nuclear images of cancellative ones.

Integrally closed pomonoids are nuclear images of subgroups of ordered groups.

Syntactic characterization of quasivarieties closed under nuclear images.

Abstract

Two fundamental constructions operating on residuated lattices and partially ordered monoids (pomonoids) are so-called nuclear images and conuclear images. Nuclear images allow us to construct many of the ordered algebras which arise in non-classical logic (such as pomonoids, semilattice-ordered monoids, and residuated lattices) from cancellative ones. Conuclear images then allow us to construct some of these cancellative algebras from partially ordered or lattice-ordered groups. Among other things, we show that finite (commutative) integral residuated lattices are precisely the finite nuclear images of commutative cancellative integral residuated lattices and that (commutative) integrally closed pomonoids are precisely the nuclear images of subpomonoids of partially ordered (Abelian) groups. The key construction is the free nuclear preimage of a pomonoid. As a by-product of our study of free nuclear preimages, we obtain a syntactic characterization of quasivarieties of pomonoids and semilattice-ordered monoids closed under nuclear images.