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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bahls, P., J. Cole, N. Galatos, P. Jipsen, and C. Tsinakis, Cancellative residuated lattices, Algebr. Univ. 50(1):83–106, 2003.
Balbes, R., and Ph. Dwinger, Distributive Lattices, University of Missouri Press, Columbia, MO., 1974.
Blok, W. J., and C. J. Van Alten, On the finite embeddability property for residuated ordered groupoids, Trans. Amer. Math. Soc. 357(10):4141–4157, 2005.
Blount, K., and C. Tsinakis, The structure of residuated lattices, Int. J. Algebr. Comput. 13(4):437–461, 2003.
Blyth, T. S., Lattices and Ordered Algebraic Structures, Universitext, Springer-Verlag London, Ltd., London, 2005.
Casari, E., Comparative logics, Synthese 73(3):421–449, 1987.
Casari, E., Comparative logics and abelian \(\ell \)-groups, in C. Bonotto, R. Ferro, S. Valentini, and A. Zanardo, (eds.), Logic Colloquium ’88, Elsevier, 1989, pp. 161–190.
Casari, E., Conjoining and disjoining on different levels, in M. L. Dalla Chiara, (ed.), Logic and Scientific Methods, Kluwer, Dordrecht, 1997, pp. 261–288.
Ciabattoni, A., N. Galatos, and K. Terui, Algebraic proof theory for substructural logics: Cut-elimination and completions, Ann. Pure Appl. Logic 163(3):266–290, 2012.
Clay, A., and D. Rolfsen, Ordered Groups and Topology, vol. 176 of Graduate Studies in Mathematics, American Mathematical Society, 2016.
Dudek, W. A., and Y. B. Jun, Pseudo-BCI algebras, East Asian Math. J. 24(2):187–190, 2008.
Emanovský, P., and J. Kühr, Some properties of pseudo-BCK- and pseudo-BCI-algebras, Fuzzy Sets and Systems 339:1–16, 2018.
Fuchs, L., Partially Ordered Algebraic Systems, Pergamon Press, Oxford, 1963.
Galatos, N., and P. Jipsen, Residuated frames with applications to decidability, Trans. Amer. Math. Soc. 364:1219–1249, 2013.
Galatos, N., P. Jipsen, T. Kowalski, and H. Ono, Residuated Lattices: An Algebraic Glimpse at Substructural Logics, Elsevier, Amsterdam, 2007.
Galatos, N., and G. Metcalfe, Proof theory for lattice-ordered groups, Ann. Pure Appl. Logic 8(167):707–724, 2016.
Galatos, N., and H. Ono, Glivenko theorems for substructural logics over FL, J. Symbolic Logic 71(4):1353–1384, 2006.
Galatos, N., and C. Tsinakis, Generalized MV-algebras, J. Algebra 283(1):254–291, 2005.
Glass, A. M. W., and Y. Gurevich, The word problem for lattice-ordered groups, Trans. Amer. Math. Soc. 280(1):127–138, 1983.
Holland, W. C., and S. H. McCleary, Solvability of the word problem in free lattice-ordered groups, Houston J. Math. 5(1):99–105, 1979.
Horčík, R., and K. Terui, Disjunction property and complexity of substructural logics, Theoret. Comput. Sci. 412(31):3992–4006, 2011.
Iséki, K., An algebra related with a propositional calculus, Proc. Japan Acad. 42:26–29, 1966.
Jipsen, P., and C. Tsinakis, A survey of residuated lattices, in J. Martinez, (ed.), Ordered Algebraic Structures, Kluwer, Alphen aan den Rijn, 2002, pp. 19–56.
Kashima, R., and Y. Komori, The word problem for free BCI-algebras is decidable, Math. Japon. 37(6):1025–1029, 1992.
Kowalski, T., and S. Butchart, A note on monothetic BCI, Notre Dame J. Formal Logic 47(4):541–544, 2006.
Kühr, J., Pseudo BCK-algebras and residuated lattices, in Contributions to General Algebra 16, Heyn, Klagenfurt, 2005, pp. 139–144.
Metcalfe, G., Proof calculi for Casari’s comparative logics, J. Logic Comput. 16(4):405–422, 2006.
Metcalfe, G., F. Paoli, and C. Tsinakis, Ordered algebras and logic, in H. Hosni, and F. Montagna, (eds.), Uncertainty and Rationality, Publications of the Scuola Normale Superiore di Pisa, Vol. 10, 2010, pp. 1–85.
Montagna, F., and C. Tsinakis, Ordered groups with a conucleus, J. Pure Appl. Algebra 214(1):71–88, 2010.
Paoli, F., The proof theory of comparative logic, Logique et Analyse 171:357–370, 2000.
Raftery, J. G., and C. J. van Alten, Residuation in commutative ordered monoids with minimal zero, Rep. Math. Logic 34:23–57, 2000.
Savitch, W. J., Relationships between nondeterministic and deterministic tape complexities, J. Comput. Syst. Sci. 4(2):177–192, 1970.
Acknowledgements
The research reported in this paper was supported by Swiss National Science Foundation (SNF) Grant 200021_184693.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
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
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11225-019-09888-9