×
Rachůnek, Jiří; Švrček, Filip

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

Order 25, No. 2, 157-175 (2008).
The main goal of this paper is to introduce and study monadic residuated \(l\)-monoids as a generalization of monadic MV-algebras.
Reviewer: Marius Tarnauceanu (Iaşi)

Cited in 13 Documents

MSC:

06F05 Ordered semigroups and monoids
03G25 Other algebras related to logic
06D20 Heyting algebras (lattice-theoretic aspects)
06D35 MV-algebras

Keywords:

residuated \(l\)-monoid; Heyting algebra; universal quantifier
PDF BibTeX XML Cite
\textit{J. Rachůnek} and \textit{F. Švrček}, Order 25, No. 2, 157--175 (2008; Zbl 1151.06008)
Full Text: DOI
  • logo of hbz
  • logo of WorldCat

References:

[1] Bahls, P., Cole, J., Galatos, N., Jipsen, P., Tsinakis, C.: Cancellative residuated lattices. Algebra Univers. 50, 83–106 (2003) · Zbl 1092.06012
[2] Belluce, L.P., Grigolia, R., Lettieri, A.: Representations of monadic MV-algebras. Stud. Log. 81, 123–144 (2005) · Zbl 1093.06008
[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
[5] Bezhanisvili, G.: Varieties of monadic Heyting algebras II. Stud. Log. 62, 21–48 (1999) · Zbl 0973.06010
[6] Bezhanisvili, G.: Varieties of monadic Heyting algebras III. Stud. Log. 64, 215–256 (2000) · Zbl 0980.06007
[7] Bezhanisvili, G., Harding, J.: Functional monadic Heyting algebras. Algebra Univers. 48, 1–10 (2002) · Zbl 1062.06017
[8] Blount, K., Tsinakis, C.: The structure of residuated lattices. Int. J. Algebra Comput. 13, 437–461 (2003) · Zbl 1048.06010
[9] Di Nola, A., Grigolia, R.: On monadic MV-algebras. Ann. Pure Appl. Logic 128, 125–139 (2004) · Zbl 1052.06010
[10] Dvurečenskij, A., Rachunek, J.: Bounded commutative residuated l-monoids with general comparability and states. Soft Comput. 10, 212–218 (2006) · Zbl 1096.06008
[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
[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
[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
[29] Rachunek, J., Šalounová, D.: Local bounded commutative residuated l-monoids. Czechoslov. Math. J. 57, 395–406 (2007) · Zbl 1174.06331
[30] Rachunek, J., Šalounová, D.: Truth values on generalizations of some commutative fuzzy structures. Fuzzy Sets Syst. 157, 3159–3168 (2006) · Zbl 1114.03021
[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
[33] Schwartz, D.: Polyadic MV-algebras. Zeit. f. math. Logik und Grundlagen d. Math. 26, 561–564 (1980) · Zbl 0488.03035
[34] Swamy, K.L.N.: Dually residuated lattice ordered semigroups. Math. Ann. 159, 105–114 (1965) · Zbl 0135.04203
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.