  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.
