# Article

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Keywords:
residuated lattice; bounded integral residuated lattice; interior operator; closure operator
Summary:
Commutative bounded integral residuated lattices form a large class of algebras containing some classes of algebras behind many valued and fuzzy logics. In the paper we introduce and investigate additive closure and multiplicative interior operators on this class of algebras.
References:
[1] Balbes, R., Dwinger, P.: Distributive Lattices. University Missouri Press, Columbia, 1974. MR 0373985 | Zbl 0321.06012
[2] Cignoli, R. L. O., D’Ottaviano, M. L., Mundici, D.: Algebraic Foundations of Many-valued Reasoning. Kluwer Academic Publishers, Dordrecht, 2000. MR 1786097
[3] Cignoli, R., Torrens, A.: Glivenko like theorems in natural expansions of BCK-logic. Math. Log. Quart. 50 (2004), 111–125. DOI 10.1002/malq.200310082 | MR 2037731 | Zbl 1045.03026
[4] Ciungu, L. C.: Classes of residuated lattices. Annals of University of Craiova. Math. Comp. Sci. Ser. 33 (2006), 180–207. MR 2359903 | Zbl 1119.03343
[5] Dvurečenskij, A., Rachůnek, J.: On Riečan and Bosbach states for bounded R$\ell$-monoids. Math. Slovaca 56 (2006), 487–500. MR 2293582
[6] Dvurečenskij, A., Rachůnek, J.: Probabilistic averaging in bounded commutative residuated $\ell$-monoids. Discrete Math. 306 (2006), 1317–1326. DOI 10.1016/j.disc.2005.12.024 | MR 2237716 | Zbl 1105.06011
[7] Esteva, F., Godo, L.: Monoidal t-norm based logic: towards a logic for left-continuous t-norms. Fuzzy Sets Syst. 124 (2001), 271–288. MR 1860848 | Zbl 0994.03017
[8] Galatos, N., Jipsen, P., Kowalski, T., Ono, H.: Residuated Lattices: An Algebraic Glimpse at Substructural Logics. Elsevier, Amsterdam, 2007. MR 2531579 | Zbl 1171.03001
[9] Hájek, P.: Metamathematics of Fuzzy Logic. Springer, Dordrecht, 1998. MR 1900263
[10] Jipsen, P., Montagna, A.: The Blok-Ferreirim theorem for normal GBL-algebras and its application. Algebra Universalis 60 (2009), 381–404. DOI 10.1007/s00012-009-2106-4 | MR 2504748 | Zbl 1192.06011
[11] Jipsen, P., Tsinakis, C.: A Survey of Residuated Lattices. In: Ordered Algebraic Structures, Kluwer, Dordrecht, (2006), 19–56. MR 2083033
[12] Rachůnek, J., Slezák, V.: Negation in bounded commutative DR$\ell$-monoids. Czechoslovak Math. J. 56 (2007), 755–763. DOI 10.1007/s10587-006-0053-1
[13] Rachůnek, J., Švrček, F.: MV-algebras with additive closure operators. Acta Univ. Palacki. Olomouc., Fac. Rer. Nat., Math. 39 (2000), 183–189. MR 1826361 | Zbl 1039.06005
[14] Rachůnek, J., Švrček, F.: Interior and closure operators on bounded commutative residuated $\ell$-monoids. Discuss. Math., Gen. Alg. Appl. 28 (2008), 11–27. DOI 10.7151/dmgaa.1132 | MR 2437765 | Zbl 1227.06014
[15] Sikorski, R.: Boolean Algebras. 2nd edition, Springer-Verlag, Berlin–Göttingen–Heidelbeg–New York, 1963. Zbl 0122.26101

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