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Canonical formulas for \(k\)-potent commutative, integral, residuated lattices. (English) Zbl 1420.03147
Summary: Canonical formulas are a powerful tool for studying intuitionistic and modal logics. Indeed, they provide a uniform and semantic way of axiomatising all extensions of intuitionistic logic and all modal logics above K4. Although the method originally hinged on the relational semantics of those logics, recently it has been completely recast in algebraic terms. In this new perspective, canonical formulas are built from a finite subdirectly irreducible algebra by describing completely the behaviour of some operations and only partially the behaviour of some others. In this paper, we export the machinery of canonical formulas to substructural logics by introducing canonical formulas for \(k\)-potent, commutative, integral, residuated lattices (\(k\)-CIRL). We show that any subvariety of \(k\)-CIRL is axiomatised by canonical formulas. The paper ends with some applications and examples.

03G25 Other algebras related to logic
03B47 Substructural logics (including relevance, entailment, linear logic, Lambek calculus, BCK and BCI logics)
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[1] Bezhanishvili, G.; Bezhanishvili, N., An algebraic approach to canonical formulas: intuitionistic case, Rev. Symb. Log., 2, 517-549, (2009) · Zbl 1183.03065
[2] Bezhanishvili, G.; Bezhanishvili, N., An algebraic approach to canonical formulas: modal case, Studia Logica, 99, 93-125, (2011) · Zbl 1250.03124
[3] Bezhanishvili, G.; Bezhanishvili, N., Canonical formulas for wk4, Rev. Symb. Log., 5, 731-762, (2012) · Zbl 1314.03020
[4] Bezhanishvili, G.; Bezhanishvili, N., Locally finite reducts of Heyting algebras and canonical formulas, Notre Dame J. Form. Log., 58, 21-45, (2017) · Zbl 1417.03198
[5] Bezhanishvili, G.; Bezhanishvili, N.; Iemhoff, R., Stable canonical rules, J. Symb. Log., 81, 284-315, (2016) · Zbl 1345.03034
[6] Bezhanishvili, N.: Lattices of intermediate and cylindric modal logics. Ph.D. thesis, University of Amsterdam (2006) · Zbl 1397.03016
[7] Bezhanishvili, N., Frame based formulas for intermediate logics, Studia Logica, 90, 139-159, (2008) · Zbl 1172.03019
[8] Bezhanishvili, N.; Gabelaia, D.; Ghilardi, S.; Jibladze, M., Admissible bases via stable canonical rules, Studia Logica, 104, 317-341, (2016) · Zbl 1397.03016
[9] Bezhanishvili, N., Ghilardi, S.: Multiple-conclusion rules, hypersequents syntax and step frames. In: Gore, R., Kooi, B., Kurucz A. (eds.) Advances in Modal Logic (AiML 2014), pp. 54-61. College Publications (2014) · Zbl 1385.03016
[10] Bezhanishvili, N., de Jongh, D.: Stable formulas in intuitionistic logic. Notre Dame J. Form. Log. (to appear) · Zbl 1256.03019
[11] Blok, W.J., Pigozzi, D.: Algebraizable logics. Mem. Amer. Math. Soc. 77 (1989) · Zbl 0664.03042
[12] Blok, W.J., Van Alten, C.J.: The finite embeddability property for residuated lattices, pocrims and BCK-algebras. Algebra Universalis 48, 253-271 (2002) · Zbl 1058.06016
[13] Burris S., Sankappanavar H.P.: A Course in Universal Algebra. Springer, New York (1981) · Zbl 0478.08001
[14] Chagrov A., Zakharyaschev M.: Modal Logic. The Clarendon Press, New York (1997) · Zbl 0871.03007
[15] Ciabattoni, A., Galatos, N., Terui, K.: From axioms to analytic rules in nonclassical logics. Proceedings of LICS’08 pp. 229-240 (2008)
[16] Ciabattoni, A.; Galatos, N.; Terui, K., Macneille completions of fl-algebras, Algebra Universalis, 66, 405-420, (2011) · Zbl 1259.03086
[17] Ciabattoni, A.; Galatos, N.; Terui, K., Algebraic proof theory for substructural logics: cut-elimination and completions, Ann. Pure Appl. Logic, 163, 266-290, (2012) · Zbl 1245.03026
[18] Citkin, A.: Characteristic formulas 50 years later (an algebraic account). ArXiv:1407.5823 [math.LO] · Zbl 1397.03016
[19] Galatos, N.; Jipsen, P., Residuated frames with applications to decidability, Trans. Amer. Math. Soc., 365, 1219-1249, (2013) · Zbl 1285.03077
[20] Galatos, N., Jipsen, P., Kowalski, T., Ono, H.: Residuated Lattices: an Algebraic Glimpse at Substructural Logics. Elsevier (2007) · Zbl 1171.03001
[21] Jankov, V., The construction of a sequence of strongly independent superintuitionistic propositional calculi, Soviet Math. Dokl., 9, 806-807, (1968) · Zbl 0198.31803
[22] Jeřábek, E., Canonical rules, J. Symb. Log., 74, 1171-1205, (2009) · Zbl 1186.03045
[23] Jeřábek, E., A note on the substructural hierarchy, Mathematical Logic Quarterly, 62, 102-110, (2016) · Zbl 1357.03057
[24] Kowalski, T.; Ono, H., Splittings in the variety of residuated lattices, Algebra Universalis, 44, 283-298, (2000) · Zbl 1015.08008
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