Classes of fuzzy filters of residuated lattice ordered monoids

Jiří Rachůnek; Dana Šalounová

Mathematica Bohemica (2010)

  • Volume: 135, Issue: 1, page 81-97
  • ISSN: 0862-7959

Abstract

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The logical foundations of processes handling uncertainty in information use some classes of algebras as algebraic semantics. Bounded residuated lattice ordered monoids (monoids) are common generalizations of BL -algebras, i.e., algebras of the propositional basic fuzzy logic, and Heyting algebras, i.e., algebras of the propositional intuitionistic logic. From the point of view of uncertain information, sets of provable formulas in inference systems could be described by fuzzy filters of the corresponding algebras. In the paper we investigate implicative, positive implicative, Boolean and fantastic fuzzy filters of bounded Rl -monoids.

How to cite

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Rachůnek, Jiří, and Šalounová, Dana. "Classes of fuzzy filters of residuated lattice ordered monoids." Mathematica Bohemica 135.1 (2010): 81-97. <http://eudml.org/doc/38113>.

@article{Rachůnek2010,
abstract = {The logical foundations of processes handling uncertainty in information use some classes of algebras as algebraic semantics. Bounded residuated lattice ordered monoids (monoids) are common generalizations of $\text\{\rm BL\}$-algebras, i.e., algebras of the propositional basic fuzzy logic, and Heyting algebras, i.e., algebras of the propositional intuitionistic logic. From the point of view of uncertain information, sets of provable formulas in inference systems could be described by fuzzy filters of the corresponding algebras. In the paper we investigate implicative, positive implicative, Boolean and fantastic fuzzy filters of bounded $\text\{\rm Rl\}$-monoids.},
author = {Rachůnek, Jiří, Šalounová, Dana},
journal = {Mathematica Bohemica},
keywords = {residuated $\text\{\rm l\}$-monoid; non-classical logics; basic fuzzy logic; intuitionistic logic; filter; fuzzy filter; $\text\{\rm BL\}$-algebra; $\text\{\rm MV\}$-algebra; Heyting algebra; residuated -monoid; fuzzy filter; BL-algebra; MV-algebra; Heyting algebra},
language = {eng},
number = {1},
pages = {81-97},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Classes of fuzzy filters of residuated lattice ordered monoids},
url = {http://eudml.org/doc/38113},
volume = {135},
year = {2010},
}

TY - JOUR
AU - Rachůnek, Jiří
AU - Šalounová, Dana
TI - Classes of fuzzy filters of residuated lattice ordered monoids
JO - Mathematica Bohemica
PY - 2010
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 135
IS - 1
SP - 81
EP - 97
AB - The logical foundations of processes handling uncertainty in information use some classes of algebras as algebraic semantics. Bounded residuated lattice ordered monoids (monoids) are common generalizations of $\text{\rm BL}$-algebras, i.e., algebras of the propositional basic fuzzy logic, and Heyting algebras, i.e., algebras of the propositional intuitionistic logic. From the point of view of uncertain information, sets of provable formulas in inference systems could be described by fuzzy filters of the corresponding algebras. In the paper we investigate implicative, positive implicative, Boolean and fantastic fuzzy filters of bounded $\text{\rm Rl}$-monoids.
LA - eng
KW - residuated $\text{\rm l}$-monoid; non-classical logics; basic fuzzy logic; intuitionistic logic; filter; fuzzy filter; $\text{\rm BL}$-algebra; $\text{\rm MV}$-algebra; Heyting algebra; residuated -monoid; fuzzy filter; BL-algebra; MV-algebra; Heyting algebra
UR - http://eudml.org/doc/38113
ER -

References

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  1. Balbes, R., Dwinger, P., Distributive Lattices, Univ. of Missouri Press, Columbia, Missouri (1974). (1974) Zbl0321.06012MR0373985
  2. Blount, K., Tsinakis, C., 10.1142/S0218196703001511, Intern. J. Alg. Comp. 13 (2003), 437-461. (2003) Zbl1048.06010MR2022118DOI10.1142/S0218196703001511
  3. Cignoli, R., D'Ottaviano, I. M. L., Mundici, D., Algebraic Foundations of Many-Valued Reasoning, Kluwer, Dordrecht (2000). (2000) Zbl0937.06009
  4. Dvurečenskij, A., Rachůnek, J., 10.1016/j.disc.2005.12.024, Discrete Mathematics 306 (2006), 1317-1326. (2006) MR2237716DOI10.1016/j.disc.2005.12.024
  5. Dymek, G., 10.1007/s00500-007-0170-2, Soft Comput. 12 (2008), 365-372. (2008) Zbl1132.06006MR2437768DOI10.1007/s00500-007-0170-2
  6. Galatos, N., Jipsen, P., Kowalski, T., Ono, H., Residuated Lattices: An Algebraic Glimpse at Substructural Logics, Elsevier Studies in Logic and Foundations (2007). (2007) Zbl1171.03001MR2531579
  7. Hájek, P., Metamathematics of Fuzzy Logic, Kluwer, Dordrecht (1998). (1998) MR1900263
  8. Hájek, P., 10.1007/s005000050043, Soft Comput. 2 (1998), 124-128. (1998) DOI10.1007/s005000050043
  9. Haveshki, M., Saeid, A. B., Eslami, E., 10.1007/s00500-005-0534-4, Soft Comput. 10 (2006), 657-664. (2006) Zbl1103.03062MR2402626DOI10.1007/s00500-005-0534-4
  10. Höhle, U., Commutative, residuated l-monoids, Non-Classical Logics and Their Applications to Fuzzy Subsets, Kluwer, Dordrecht U. Höhle, E. P. Klement (1995), 53-106. (1995) MR1345641
  11. Hoo, C. S., Fuzzy ideals of BCI and MV-algebras, Fuzzy Sets and Syst. 62 (1994), 111-114. (1994) Zbl0826.06011MR1259890
  12. Hoo, C. S., Fuzzy implicative and Boolean ideals of MV-algebras, Fuzzy Sets and Syst. 66 (1994), 315-327. (1994) Zbl0844.06007MR1300288
  13. Jipsen, P., Tsinakis, C., A survey of residuated lattices, J. Martinez Ordered algebraic structures. Kluwer, Dordrecht (2002), 19-56. (2002) Zbl1070.06005MR2083033
  14. Jun, Y. B., Walendziak, A., Fuzzy ideals of pseudo MV-algebras, Inter. Rev. Fuzzy Math. 1 (2006), 21-31. (2006) Zbl1128.06005MR2294714
  15. Kondo, M., Dudek, W. A., 10.1007/s00500-007-0178-7, Soft Comput. 12 (2008), 419-423. (2008) Zbl1165.03056DOI10.1007/s00500-007-0178-7
  16. Kondo, M., Dudek, W. A., On transfer principle in fuzzy theory, Mathware and Soft Comput. 13 (2005), 41-55. (2005) MR2160344
  17. Rachůnek, J., A duality between algebras of basic logic and bounded representable DRl-monoids, Math. Bohem. 126 (2001), 561-569. (2001) MR1970259
  18. Rachůnek, J., Šalounová, D., Boolean deductive systems of bounded commutative residuated l-monoids, Contrib. Gen. Algebra 16 (2005), 199-208. (2005) MR2166959
  19. Rachůnek, J., Šalounová, D., 10.1007/s10587-007-0068-2, Czech. Math. J. 57 (2007), 395-406. (2007) MR2309973DOI10.1007/s10587-007-0068-2
  20. Rachůnek, J., Šalounová, D., Classes of filters in generalizations of commutative fuzzy structures, Acta Univ. Palacki. Olomuc., Fac. Rer. Nat., Math (to appear). MR2641951
  21. Rachůnek, J., Šalounová, D., 10.1016/j.ins.2008.05.005, Inform. Sci. 178 (2008), 3474-3481. (2008) MR2436416DOI10.1016/j.ins.2008.05.005
  22. Rachůnek, J., Slezák, V., 10.1007/s10587-006-0053-1, Czech. Math. J. 56 (2006), 755-763. (2006) MR2291772DOI10.1007/s10587-006-0053-1
  23. Turunen, E., 10.1007/s001530100088, Arch. Math. Logic 40 (2001), 467-473. (2001) Zbl1030.03048MR1854896DOI10.1007/s001530100088
  24. Ward, M., Dilworth, R. P., 10.1090/S0002-9947-1939-1501995-3, Trans. Amer. Math. Soc. 45 (1939), 335-354 5.0084.01. (1939) Zbl0021.10801MR1501995DOI10.1090/S0002-9947-1939-1501995-3
  25. Zhan, J., Dudek, W. A., Jun, Y. B., 10.1007/s00500-008-0288-x, Soft Comput. 13 (2009), 13-21. (2009) MR2603340DOI10.1007/s00500-008-0288-x

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