Integrally Closed Residuated Lattices

Studia Logica 108 (5):1063-1086 (2020)

Abstract

A residuated lattice is said to be integrally closed if it satisfies the quasiequations \ and \, or equivalently, the equations \ and \. 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 \\backslash {\mathrm {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 BCI-algebras, semi-integral residuated pomonoids, and Casari’s comparative logic.

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