Abstract
A residuated lattice is said to be integrally closed if it satisfies the quasiequations \(xy \le x \implies y \le {\mathrm {e}}\) and \(yx \le ~x \implies y \le {\mathrm {e}}\), or equivalently, the equations \(x \backslash x \approx {\mathrm {e}}\) and \(x /x \approx {\mathrm {e}}\). 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 \(a \mapsto (a \backslash {\mathrm {e}})\backslash {\mathrm {e}}\) 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 (pseudo) BCI-algebras, semi-integral residuated pomonoids, and Casari’s comparative logic.
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The research reported in this paper was supported by Swiss National Science Foundation (SNF) Grant 200021_184693.
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Gil-Férez, J., Lauridsen, F.M. & Metcalfe, G. Integrally Closed Residuated Lattices. Stud Logica 108, 1063–1086 (2020). https://doi.org/10.1007/s11225-019-09888-9
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DOI: https://doi.org/10.1007/s11225-019-09888-9