Interior and closure operators on bounded residuated lattices

Jiří Rachůnek; Zdeněk Svoboda

Open Mathematics (2014)

  • Volume: 12, Issue: 3, page 534-544
  • ISSN: 2391-5455

Abstract

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Bounded integral residuated lattices form a large class of algebras containing some classes of algebras behind many valued and fuzzy logics. In the paper we introduce and investigate multiplicative interior and additive closure operators (mi- and ac-operators) generalizing topological interior and closure operators on such algebras. We describe connections between mi- and ac-operators, and for residuated lattices with Glivenko property we give connections between operators on them and on the residuated lattices of their regular elements.

How to cite

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Jiří Rachůnek, and Zdeněk Svoboda. "Interior and closure operators on bounded residuated lattices." Open Mathematics 12.3 (2014): 534-544. <http://eudml.org/doc/269666>.

@article{JiříRachůnek2014,
abstract = {Bounded integral residuated lattices form a large class of algebras containing some classes of algebras behind many valued and fuzzy logics. In the paper we introduce and investigate multiplicative interior and additive closure operators (mi- and ac-operators) generalizing topological interior and closure operators on such algebras. We describe connections between mi- and ac-operators, and for residuated lattices with Glivenko property we give connections between operators on them and on the residuated lattices of their regular elements.},
author = {Jiří Rachůnek, Zdeněk Svoboda},
journal = {Open Mathematics},
keywords = {Residuated lattice; Bounded integral residuated lattice; Interior operator; Closure operator; residuated lattice; bounded integral residuated lattice; interior operator; closure operator},
language = {eng},
number = {3},
pages = {534-544},
title = {Interior and closure operators on bounded residuated lattices},
url = {http://eudml.org/doc/269666},
volume = {12},
year = {2014},
}

TY - JOUR
AU - Jiří Rachůnek
AU - Zdeněk Svoboda
TI - Interior and closure operators on bounded residuated lattices
JO - Open Mathematics
PY - 2014
VL - 12
IS - 3
SP - 534
EP - 544
AB - Bounded integral residuated lattices form a large class of algebras containing some classes of algebras behind many valued and fuzzy logics. In the paper we introduce and investigate multiplicative interior and additive closure operators (mi- and ac-operators) generalizing topological interior and closure operators on such algebras. We describe connections between mi- and ac-operators, and for residuated lattices with Glivenko property we give connections between operators on them and on the residuated lattices of their regular elements.
LA - eng
KW - Residuated lattice; Bounded integral residuated lattice; Interior operator; Closure operator; residuated lattice; bounded integral residuated lattice; interior operator; closure operator
UR - http://eudml.org/doc/269666
ER -

References

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  1. [1] Balbes R., Dwinger P., Distributive Lattices, University of Missouri Press, Columbia, 1974 
  2. [2] Cignoli R.L.O., D’Ottaviano I.M.L., Mundici D., Algebraic Foundations of Many-Valued Reasoning, Trends Log. Stud. Log. Libr., 7, Kluwer, Dordrecht, 2000 http://dx.doi.org/10.1007/978-94-015-9480-6[Crossref] 
  3. [3] Cignoli, R., Torrens Torrell A., Glivenko like theorems in natural expansions of BCK-logic, MLQ Math. Log. Q., 2004, 50(2), 111–125 http://dx.doi.org/10.1002/malq.200310082[Crossref] Zbl1045.03026
  4. [4] Ciungu L.C., Classes of residuated lattices, An. Univ. Craiova Ser. Mat. Inform., 2006, 33, 189–207 Zbl1119.03343
  5. [5] Dvurečenskij A., Every linear pseudo BL-algebra admits a state, Soft Computing, 2007, 11(6), 495–501 http://dx.doi.org/10.1007/s00500-006-0078-2[Crossref][WoS] Zbl1122.06012
  6. [6] Dvurečenskij A., Rachůnek J., On Riečan and Bosbach states for bounded non-commutative Rℓ-monoids, Math. Slovaca, 2006, 56(5), 487–500 Zbl1141.06005
  7. [7] Dvurečenskij A., Rachůnek J., Probabilistic averaging in bounded commutative residuated ℓ-monoids, Discrete Math., 2006, 306(13), 1317–1326 http://dx.doi.org/10.1016/j.disc.2005.12.024[Crossref] Zbl1105.06011
  8. [8] Dvurečenskij A., Rachůnek J., Probabilistic averaging in bounded Rℓ-monoids, Semigroup Forum, 2006, 72(2), 190–206 
  9. [9] Esteva F., Godo L., Monoidal t-norm based logic: towards a logic for left-continuous t-norms, Fuzzy Sets and Systems, 2001, 124(3), 271–288 http://dx.doi.org/10.1016/S0165-0114(01)00098-7[Crossref] Zbl0994.03017
  10. [10] Flondor P., Georgescu G., Iorgulescu A., Pseudo-t-norms and pseudo-BL algebras, Soft Computing, 2001, 5(5), 355–371 http://dx.doi.org/10.1007/s005000100137[Crossref][WoS] Zbl0995.03048
  11. [11] Galatos N., Jipsen P., Kowalski T., Ono H., Residuated Lattices: An Algebraic Glimpse at Substructural Logics, Stud. Logic Found. Math., 151, Elsevier, Amsterdam, 2007 Zbl1171.03001
  12. [12] Georgescu G., Iorgulescu A., Pseudo-MV algebras, Mult.-Valued Logic, 2001, 6(1–2), 95–135 Zbl1014.06008
  13. [13] Hájek P., Metamathematics of Fuzzy Logic, Trends Log. Stud. Log. Libr., 4, Kluwer, Dordrecht, 1998 http://dx.doi.org/10.1007/978-94-011-5300-3[Crossref] 
  14. [14] Jipsen P., Tsinakis C., A Survey of Residuated Lattices, In: Ordered Algebraic Structures, Gainesville, February 28–March 3, 2001, Dev. Math., 7, Kluwer, Dordrecht, 2006, 19–56 Zbl1070.06005
  15. [15] di Nola A., Georgescu G., Iorgulescu A., Pseudo-BL algebras I, Mult.-Valued Log., 2002, 8(5–6), 673–714 Zbl1028.06007
  16. [16] Rachůnek J., A non-commutative generalization of MV-algebras, Czechoslovak Math. J., 2002, 52(2), 255–273 http://dx.doi.org/10.1023/A:1021766309509[Crossref] Zbl1012.06012
  17. [17] Rachůnek J., Šalounová D., A generalization of local fuzzy structures, Soft Computing, 2007, 11(6), 565–571 http://dx.doi.org/10.1007/s00500-006-0101-7[WoS][Crossref] Zbl1121.06013
  18. [18] Rachůnek J., Šalounová D., States on Generalizations of Fuzzy Structures, Palacký University Press, Olomouc, 2011 
  19. [19] Rachůnek J., Slezák V., Negation in bounded commutative DRℓ-monoids, Czechoslovak Math. J., 2007, 56(131)(2), 755–763 
  20. [20] Rachůnek J., Švrček F., MV-algebras with additive closure operators, Acta Univ. Palack. Olomouc. Fac. Rerum Natur. Math., 2000, 39, 183–189 
  21. [21] Rachůnek J., Švrček F., Interior and closure operators on bounded commutative residuated ℓ-monoids, Discuss. Math. Gen. Algebra Appl., 2008, 28(1), 11–27 http://dx.doi.org/10.7151/dmgaa.1132[Crossref] 
  22. [22] Sikorski R., Boolean Algebras, 2nd ed., Ergeb. Math. Grenzgeb., 25, Academic Press, New York/Springer, Berlin-New York, 1964 
  23. [23] Švrček F., Interior and closure operators on bounded residuated lattice ordered monoids, Czechoslovak Math. J., 2008, 58(133) (2), 345–357 [WoS] Zbl1174.06323

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