An Investigation of Residuated Lattices with Modal Operators

Abstract

Residuated lattices, which generalize Boolean algebras and lattice-ordered groups, have been useful in the study of algebraic logic, particularly as an algebraic semantics for substructural logics. By equipping a residuated lattice with a modal operator (either a nucleus or a conucleus) and then considering the image under this operator, a new class of residuated lattices results. For example, Heyting algebras arise as the conuclear images of Boolean algebras; commutative, cancellative residuated lattices as the conuclear images of Abelian lattice-ordered groups; and integral GMV-algebras as the nuclear images of negative cones of lattice-ordered groups.

In all three of the aforementioned cases, there is a categorical equivalence when you restrict to a certain subcategory of the class of residuated lattices with modal operators. Also, there is a strong connection between the subvariety lattice of the class of residuated lattices with modal operators and the corresponding class of their images under this operator. In particular, we show an even stronger connection in the case of negatively-pointed Abelian lattice-ordered groups (which can be seen as a residuated lattice with the nucleus that forms the interval from the designated negative element to the identity) and their images (which are MV-algebras). Namely, the subvariety lattice of negatively-pointed Abelian lattice-ordered groups (without the trivial variety) is isomorphic to the subvariety lattice of MV-algebras.