Article
Full entry |
PDF
(0.3 MB)
Feedback
PDF
(0.3 MB)
Feedback
Keywords:
residuated $\ell $-monoid; residuated lattice; $BL$-algebra; $MV$-algebra; local $R\ell $-monoid; filter
residuated $\ell $-monoid; residuated lattice; $BL$-algebra; $MV$-algebra; local $R\ell $-monoid; filter
Summary:
Bounded commutative residuated lattice ordered monoids ($R\ell $-monoids) are a common generalization of, e.g., $BL$-algebras and Heyting algebras. In the paper, the properties of local and perfect bounded commutative $R\ell $-monoids are investigated.
References:
[1] P. Bahls, J. Cole, N. Galatos, P. Jipsen, and C. Tsinakis: Cancellative residuated lattices. Alg. Univ. 50 (2003), 83–106. DOI 10.1007/s00012-003-1822-4 | MR 2026830
[2] K. Blount, C. Tsinakis: The structure of residuated lattices. Intern. J. Alg. Comp. 13 (2003), 437–461. DOI 10.1142/S0218196703001511 | MR 2022118
[3] R. L. O. Cignoli, I. M. L. D’Ottaviano, and D. Mundici: Algebraic Foundations of Many-Valued Reasoning. Kluwer Acad. Publ., Dordrecht-Boston-London, 2000. MR 1786097
[4] A. Dvurečenskij, S. Pulmannová: New Trends in Quantum Structures. Kluwer Acad. Publ., Dordrecht-Boston-London, 2000. MR 1861369
[5] A. Dvurečenskij, J. Rachůnek: Probabilistic averaging in bounded residuated $\ell $-monoids. Semigroup Forum. 72 (2006), 191–206. DOI 10.1007/s00233-005-0545-6 | MR 2216089
[6] A. Dvurečenskij, J. Rachůnek: Bounded commutative residuated $\ell $-monoids with general comparability and states. Soft Comput. 10 (2006), 212–218. DOI 10.1007/s00500-005-0473-0
[7] P. Hájek: Metamathematics of Fuzzy Logic. Kluwer, Amsterdam, 1998. MR 1900263
[8] P. Jipsen, C. Tsinakis: A survey of residuated lattices. In: Ordered Algebraic Structures, J. Martinez (ed.), Kluwer Acad. Publ., Dordrecht, 2002, pp. 19–56. MR 2083033
[9] J. Rachůnek: $DR\ell $-semigroups and $MV$-algebras. Czechoslovak Math. J. 48 (1998), 365–372. DOI 10.1023/A:1022801907138 | MR 1624268
[10] J. Rachůnek: $MV$-algebras are categorically equivalent to a class of $DR\ell _{1(i)}$ semigroups. Math. Bohemica 123 (1998), 437–441. MR 1667115
[11] J. Rachůnek: A duality between algebras of basic logic and bounded representable $DR\ell $-monoids. Math. Bohemica 126 (2001), 561–569. MR 1970259
[12] J. Rachůnek, D. Šalounová: Boolean deductive systems of bounded commutative residuated $\ell $-monoids. Contrib. Gen. Algebra 16 (2005), 199–207. MR 2166959
[13] J. Rachůnek, V. Slezák: Negation in bounded commutative $DR\ell $-monoids. Czechoslovak Math. J. 56 (2006), 755–763. DOI 10.1007/s10587-006-0053-1 | MR 2291772
[14] J. Rachůnek, V. Slezák: Bounded dually residuated lattice ordered monoids as a generalization of fuzzy structures. Math. Slovaca. 56 (2006), 223–233. MR 2229343
[15] K. L. N. Swamy: Dually residuated lattice ordered semigroups. Math. Ann. 159 (1965), 105–114. DOI 10.1007/BF01360284 | MR 0183797 | Zbl 0138.02104
[16] K. L. N. Swamy: Dually residuated lattice ordered semigroups III. Math. Ann. 167 (1966), 71–74. DOI 10.1007/BF01361218 | MR 0200364 | Zbl 0158.02601
[17] E. Turunen: Mathematics Behind Fuzzy Logic. Physica-Verlag, Heidelberg-New York, 1999. MR 1716958 | Zbl 0940.03029
[18] E. Turunen, S. Sessa: Local $BL$-algebras. Multip. Val. Logic 6 (2001), 229–250. MR 1817445
Search
Partner of
