Integrally Closed Residuated Lattices

TL;DR

This paper introduces integrally closed residuated lattices, explores their properties, and establishes a Glivenko-style correspondence with lattice-ordered groups, leading to a decidable and PSPACE-complete equational theory.

Contribution

It characterizes integrally closed residuated lattices, proves their connection to lattice-ordered groups, and develops a cut-elimination calculus with decidability results.

Findings

Mapping a -> (a\e)\e is a homomorphism onto a lattice-ordered group

Integrally closed residuated lattices form the largest variety with the Glivenko property

The equational theory is PSPACE-complete

Abstract

A residuated lattice is defined to be integrally closed if it satisfies the equations x\x = e and x/x = e. Every integral, cancellative, or divisible residuated lattice is integrally closed, and, conversely, every bounded integrally closed residuated lattice is integral. It is proved that the mapping a -> (a\e)\e on any integrally closed residuated lattice is a homomorphism onto a lattice-ordered group. A Glivenko-style property is then established for varieties of integrally closed residuated lattices with respect to varieties of lattice-ordered groups, showing in particular that integrally closed residuated lattices form the largest variety of residuated lattices admitting this property with respect to lattice-ordered groups. The Glivenko property is used to obtain a sequent calculus admitting cut-elimination for the variety of integrally closed residuated lattices and to establish the decidability, indeed PSPACE-completenes, of its equational theory. Finally, these results are related to previous work on (pseudo) BCI-algebras, semi-integral residuated partially ordered monoids, and algebras for Casari's comparative logic.