Title:
|
Monotone modal operators on bounded integral residuated lattices (English) |
Author:
|
Rachůnek, Jiří |
Author:
|
Svoboda, Zdeněk |
Language:
|
English |
Journal:
|
Mathematica Bohemica |
ISSN:
|
0862-7959 (print) |
ISSN:
|
2464-7136 (online) |
Volume:
|
137 |
Issue:
|
3 |
Year:
|
2012 |
Pages:
|
333-345 |
Summary lang:
|
English |
. |
Category:
|
math |
. |
Summary:
|
Bounded integral residuated lattices form a large class of algebras containing some classes of commutative and noncommutative algebras behind many-valued and fuzzy logics. In the paper, monotone modal operators (special cases of closure operators) are introduced and studied. (English) |
Keyword:
|
residuated lattice |
Keyword:
|
bounded integral residuated lattice |
Keyword:
|
modal operator |
Keyword:
|
closure operator |
MSC:
|
03G25 |
MSC:
|
06D35 |
MSC:
|
06F05 |
idZBL:
|
Zbl 1265.03085 |
idMR:
|
MR3112491 |
DOI:
|
10.21136/MB.2012.142898 |
. |
Date available:
|
2012-08-19T21:29:52Z |
Last updated:
|
2020-07-29 |
Stable URL:
|
http://hdl.handle.net/10338.dmlcz/142898 |
. |
Reference:
|
[1] Bahls, P., Cole, J., Galatos, N., Jipsen, P., Tsinakis, C.: Cancellative residuated lattices.Algebra Univers. 50 (2003), 83-106. Zbl 1092.06012, MR 2026830 |
Reference:
|
[2] Balbes, R., Dwinger, P.: Distributive Lattices.University Missouri Press, Columbia (1974). Zbl 0321.06012, MR 0373985 |
Reference:
|
[3] Cignoli, R. L. O., D'Ottaviano, I. M. L., Mundici, D.: Algebraic Foundations of Many-Valued Reasoning.Kluwer, Dordrecht (2000). Zbl 0937.06009 |
Reference:
|
[4] Ciungu, L. C.: Classes of residuated lattices, Annals of University of Craiova.Math. Comp. Sci. Ser. 33 (2006), 180-207. MR 2359903 |
Reference:
|
[5] DiNola, A., Georgescu, G., Iorgulesu, A.: Psedo-BL algebras; Part I.Multiple Val. Logic 8 (2002), 673-714. MR 1948853 |
Reference:
|
[6] Dowker, C. H., Papert, D.: Quotient Frames and Subspaces.Proc. London Math. Soc. 16 (1966), 275-296. Zbl 0136.43405, MR 0202648 |
Reference:
|
[7] Dvurečenskij, A.: Every linear pseudo BL-algebra admits a state.Soft Comput. 11 (2007), 495-501. Zbl 1122.06012, 10.1007/s00500-006-0078-2 |
Reference:
|
[8] Dvurečenskij, A., Rachůnek, J.: On Riečan and Bosbach states for bounded Rl-monoids.Math. Slovaca 56 (2006), 487-500. MR 2293582 |
Reference:
|
[9] Dvurečenskij, A., Rachůnek, J.: Probabilistic averaging in bounded commutative residuated l-monoids.Discrete Math. 306 (2006), 1317-1326. MR 2237716, 10.1016/j.disc.2005.12.024 |
Reference:
|
[10] Dvurečenskij, A., Rachůnek, J.: Probabilistic averaging in bounded Rl-monoids.Semigroup Forum 72 (2006), 191-206. MR 2216089, 10.1007/s00233-005-0545-6 |
Reference:
|
[11] Esteva, F., Godo, L.: Monoidal t-norm based logic: towards a logic for left-continuous t-norms.Fuzzy Sets Syst. 124 (2001), 271-288. Zbl 0994.03017, MR 1860848 |
Reference:
|
[12] Flondor, P., Georgescu, G., Iorgulescu, A.: Pseudo-t-norms and pseudo-BL algebras.Soft Comput. 5 (2001), 355-371. Zbl 0995.03048, MR 1948853, 10.1007/s005000100137 |
Reference:
|
[13] Freyd, P. J.: Aspects of topoi.Bull. Austral. Math. Soc. 7 (1972), 1-76. Zbl 0252.18002, MR 0396714, 10.1017/S0004972700044828 |
Reference:
|
[14] Galatos, N., Jipsen, P., Kowalski, T., Ono, H.: Residuated Lattices: An Algebraic Glimpse at Substructural Logics.Elsevier, Amsterdam (2007). Zbl 1171.03001, MR 2531579 |
Reference:
|
[15] Georgescu, G., Iorgulescu, A.: Pseudo-$ MV$ algebras.Multiple Val. Logic 6 (2001), 95-135. Zbl 1014.06008, MR 1817439 |
Reference:
|
[16] Hájek, P.: Metamathematics of Fuzzy Logic.Kluwer, Dordrecht (1998). MR 1900263 |
Reference:
|
[17] Harlenderová, M., Rachůnek, J.: Modal operators on $ MV$-algebras.Math. Bohem. 131 (2006), 39-48. Zbl 1112.06014, MR 2211002 |
Reference:
|
[18] Jipsen, P., Tsinakis, C.: A Survey of Residuated Lattices.Ordered Algebraic Structures, Kluwer, Dordrecht (2006), 19-56. MR 2083033 |
Reference:
|
[19] Kondo, M.: Modal operators on commutative residuated lattices.Math. Slovaca 61 (2011), 1-14. Zbl 1265.03081, MR 2772354, 10.2478/s12175-010-0055-1 |
Reference:
|
[20] Lawvere, F. W.: Quantifiers and Sheaves.Actes Congr. internat. Math. 1 (1971), 329-334. Zbl 0261.18010, MR 0430021 |
Reference:
|
[21] Lawvere, F. W.: Toposes, Algebraic Geometry and Logic.Lecture Notes 274, Springer, Berlin (1972). Zbl 0233.00005, MR 0330254 |
Reference:
|
[22] Macnab, D. S.: Modal operators on Heyting algebras.Alg. Univ. 12 (1981), 5-29. Zbl 0459.06005, MR 0608645, 10.1007/BF02483860 |
Reference:
|
[23] Rachůnek, J.: A non-commutative generalization of MV-algebras.Czech. Math. J. 52 (2002), 255-273. Zbl 1012.06012, MR 1905434, 10.1023/A:1021766309509 |
Reference:
|
[24] Rachůnek, J., Šalounová, D.: Modal operators on bounded commutative residuated l-monoids.Math. Slovaca 57 (2007), 321-332. MR 2357828, 10.2478/s12175-007-0026-3 |
Reference:
|
[25] Rachůnek, J., Šalounová, D.: A generalization of local fuzzy structures.Soft Comput. 11 (2007), 565-571. Zbl 1121.06013, 10.1007/s00500-006-0101-7 |
Reference:
|
[26] Rachůnek, J., Šalounová, D.: Modal operators on bounded residuated l-monoids.Math. Bohem. 133 (2008), 299-311. Zbl 1199.06043, MR 2494783 |
Reference:
|
[27] Rachůnek, J., Slezák, V.: Bounded dually residuated lattice ordered monoids as a generalization of fuzzy structures.Math. Slovaca 56 (2006), 223-233. Zbl 1150.06015, MR 2229343 |
Reference:
|
[28] Wraith, G. C.: Lectures on elementary topoi.Model Theor. Topoi, Collect. Lect. var. Auth., Lect. Notes Math. 445 (1975), 114-206. Zbl 0323.18005, MR 0393179, 10.1007/BFb0061296 |
. |