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Monadic bounded commutative residuated \(\ell\)-monoids. (English) Zbl 1151.06008

The main goal of this paper is to introduce and study monadic residuated \(l\)-monoids as a generalization of monadic MV-algebras.

MSC:

06F05 Ordered semigroups and monoids
03G25 Other algebras related to logic
06D20 Heyting algebras (lattice-theoretic aspects)
06D35 MV-algebras
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[1] Bahls, P., Cole, J., Galatos, N., Jipsen, P., Tsinakis, C.: Cancellative residuated lattices. Algebra Univers. 50, 83–106 (2003) · Zbl 1092.06012 · doi:10.1007/s00012-003-1822-4
[2] Belluce, L.P., Grigolia, R., Lettieri, A.: Representations of monadic MV-algebras. Stud. Log. 81, 123–144 (2005) · Zbl 1093.06008 · doi:10.1007/s11225-005-2805-6
[3] Bělohlávek, R.: Fuzzy Relational Systems. Foundations and Principles. Kluwer, New York (2002)
[4] Bezhanisvili, G.: Varieties of monadic Heyting algebras I. Stud. Log. 61, 367–402 (1998) · Zbl 0964.06008 · doi:10.1023/A:1005073905902
[5] Bezhanisvili, G.: Varieties of monadic Heyting algebras II. Stud. Log. 62, 21–48 (1999) · Zbl 0973.06010 · doi:10.1023/A:1005173628262
[6] Bezhanisvili, G.: Varieties of monadic Heyting algebras III. Stud. Log. 64, 215–256 (2000) · Zbl 0980.06007 · doi:10.1023/A:1005285631357
[7] Bezhanisvili, G., Harding, J.: Functional monadic Heyting algebras. Algebra Univers. 48, 1–10 (2002) · Zbl 1062.06017 · doi:10.1007/s00012-002-8202-3
[8] Blount, K., Tsinakis, C.: The structure of residuated lattices. Int. J. Algebra Comput. 13, 437–461 (2003) · Zbl 1048.06010 · doi:10.1142/S0218196703001511
[9] Di Nola, A., Grigolia, R.: On monadic MV-algebras. Ann. Pure Appl. Logic 128, 125–139 (2004) · Zbl 1052.06010 · doi:10.1016/j.apal.2003.11.031
[10] Dvurečenskij, A., Rachunek, J.: Bounded commutative residuated l-monoids with general comparability and states. Soft Comput. 10, 212–218 (2006) · Zbl 1096.06008 · doi:10.1007/s00500-005-0473-0
[11] Fisher Servi, G.: Axiomatizations for some intuitionistic modal logics. Rend. Semin. Mat. Univ. Polit. 42, 179–194 (1984) · Zbl 0592.03011
[12] Galatos, N., Jipsen, P., Kowalski, T., Ono, H.: Residuated Lattices: An Algebraic Glimpse at Substructual Logics, Elsevier Studies in Logic and Foundations. Elsevier, New York (2007) · Zbl 1171.03001
[13] Georgescu, G., Iorgulescu, A., Leustean, I.: Monadic and closure MV-algebras. Multi. Val. Logic. 3, 235–257 (1998) · Zbl 0920.06004
[14] Hájek, P.: Metamathematics of Fuzzy Logic. Kluwer, Dordrecht (1998) · Zbl 0937.03030
[15] Halmos, P.R.: Algebraic Logic. Chelsea, New York (1962) · Zbl 0101.01101
[16] Henkin, L., Monk, J.D., Tarski, A.: Cylindric Algebras, Part I. North-Holland, Amsterdam (1971) · Zbl 0214.01302
[17] Henkin, L., Monk, J.D., Tarski, A.: Cylindric Algebras, Part II. North-Holland, Amsterdam (1985) · Zbl 0576.03043
[18] Jipsen, P., Tsinakis, C.: A survey of residuated lattices. In: Martinez, J. (ed.) Ordered Algebraic Structures, pp. 19–56. Kluwer, Dordrecht (2002) · Zbl 1070.06005
[19] Kühr, J.: Pseudo BL-algebras and DR-monoids. Math. Bohem. 128, 199–208 (2003) · Zbl 1024.06005
[20] Monteiro, A.: Normalidad de las álgebras de Heyting monádicas. Actas de las X Jornades de la UMA, pp. 50 – 51. Bahia Blanca (1957)
[21] Monteiro, A., Varsavsky, O.: Algebras de Heyting monádicas. Actas de las X Jornades de la UMA, 52 – 62. Bahia Blanca (1957)
[22] Neméti, I.: Algebraization of quantifier logics. Stud. Log. 50, 485–569 (1991) · Zbl 0772.03033 · doi:10.1007/BF00370684
[23] Ono, H.: On Some Intuitionistic Modal Logics, vol. 13, pp. 687–722. Publications of Research Institute for Mathematical Sciences, Kyoto University. Kyoto University, Kyoto (1977) · Zbl 0373.02026
[24] Pigozzi, D., Salibra, A.: Polyadic algebras over nonclassical logics. In: Algebraic Methods in Logic and in Computer Science, vol. 28, pp. 51–66. Banach Center, Warsaw (1993) · Zbl 0794.03092
[25] Rachunek, J.: DRl-semigroups and MV-algebras. Czechoslov. Math. J. 48, 365–372 (1998) · Zbl 0952.06014 · doi:10.1023/A:1022801907138
[26] Rachunek, J.: MV-algebras are categorically equivalent to a class of DRl 1(i)-semigroups. Math. Bohem. 123, 437–441 (1998) · Zbl 0934.06014
[27] Rachunek, J.: A duality between algebras of basic logic and bounded representable DRl-monoids. Math. Bohem. 126, 561–569 (2001)
[28] Rachunek, J., Slezák, V.: Negation in bounded commutative DRl-monoids. Czechoslov. Math. J. 56, 755–763 (2006) · Zbl 1164.06325 · doi:10.1007/s10587-006-0053-1
[29] Rachunek, J., Šalounová, D.: Local bounded commutative residuated l-monoids. Czechoslov. Math. J. 57, 395–406 (2007) · Zbl 1174.06331 · doi:10.1007/s10587-007-0068-2
[30] Rachunek, J., Šalounová, D.: Truth values on generalizations of some commutative fuzzy structures. Fuzzy Sets Syst. 157, 3159–3168 (2006) · Zbl 1114.03021 · doi:10.1016/j.fss.2006.08.007
[31] Rutledge, J.D.: A preliminary investigation of the infinitely many-valued predicate calculus. Ph.D. Thesis, Cornell University (1959)
[32] Schwartz, D.: Theorie der polyadischen MV-Algebren endlicher Ordnung. Math. Nachr. 78, 131–138 (1977) · Zbl 0402.03054 · doi:10.1002/mana.19770780111
[33] Schwartz, D.: Polyadic MV-algebras. Zeit. f. math. Logik und Grundlagen d. Math. 26, 561–564 (1980) · Zbl 0488.03035 · doi:10.1002/malq.19800263602
[34] Swamy, K.L.N.: Dually residuated lattice ordered semigroups. Math. Ann. 159, 105–114 (1965) · Zbl 0135.04203 · doi:10.1007/BF01360284
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