Ultra L I -Ideals in lattice implication algebras and M T L -algebras

Xiaohong Zhang; Ke Yun Qin; Wiesław Aleksander Dudek

Czechoslovak Mathematical Journal (2007)

  • Volume: 57, Issue: 2, page 591-605
  • ISSN: 0011-4642

Abstract

top
A mistake concerning the ultra L I -ideal of a lattice implication algebra is pointed out, and some new sufficient and necessary conditions for an L I -ideal to be an ultra L I -ideal are given. Moreover, the notion of an L I -ideal is extended to M T L -algebras, the notions of a (prime, ultra, obstinate, Boolean) L I -ideal and an I L I -ideal of an M T L -algebra are introduced, some important examples are given, and the following notions are proved to be equivalent in M T L -algebra: (1) prime proper L I -ideal and Boolean L I -ideal, (2) prime proper L I -ideal and I L I -ideal, (3) proper obstinate L I -ideal, (4) ultra L I -ideal.

How to cite

top

Zhang, Xiaohong, Qin, Ke Yun, and Dudek, Wiesław Aleksander. "Ultra $LI$-Ideals in lattice implication algebras and $MTL$-algebras." Czechoslovak Mathematical Journal 57.2 (2007): 591-605. <http://eudml.org/doc/31149>.

@article{Zhang2007,
abstract = {A mistake concerning the ultra $LI$-ideal of a lattice implication algebra is pointed out, and some new sufficient and necessary conditions for an $LI$-ideal to be an ultra $LI$-ideal are given. Moreover, the notion of an $LI$-ideal is extended to $MTL$-algebras, the notions of a (prime, ultra, obstinate, Boolean) $LI$-ideal and an $ILI$-ideal of an $MTL$-algebra are introduced, some important examples are given, and the following notions are proved to be equivalent in $MTL$-algebra: (1) prime proper $LI$-ideal and Boolean $LI$-ideal, (2) prime proper $LI$-ideal and $ILI$-ideal, (3) proper obstinate $LI$-ideal, (4) ultra $LI$-ideal.},
author = {Zhang, Xiaohong, Qin, Ke Yun, Dudek, Wiesław Aleksander},
journal = {Czechoslovak Mathematical Journal},
keywords = {lattice implication algebra; $MTL$-algebra; (prime; ultra; obstinate; Boolean) $LI$-ideal; $ILI$-ideal; MTL-algebra; ILI-ideal},
language = {eng},
number = {2},
pages = {591-605},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Ultra $LI$-Ideals in lattice implication algebras and $MTL$-algebras},
url = {http://eudml.org/doc/31149},
volume = {57},
year = {2007},
}

TY - JOUR
AU - Zhang, Xiaohong
AU - Qin, Ke Yun
AU - Dudek, Wiesław Aleksander
TI - Ultra $LI$-Ideals in lattice implication algebras and $MTL$-algebras
JO - Czechoslovak Mathematical Journal
PY - 2007
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 57
IS - 2
SP - 591
EP - 605
AB - A mistake concerning the ultra $LI$-ideal of a lattice implication algebra is pointed out, and some new sufficient and necessary conditions for an $LI$-ideal to be an ultra $LI$-ideal are given. Moreover, the notion of an $LI$-ideal is extended to $MTL$-algebras, the notions of a (prime, ultra, obstinate, Boolean) $LI$-ideal and an $ILI$-ideal of an $MTL$-algebra are introduced, some important examples are given, and the following notions are proved to be equivalent in $MTL$-algebra: (1) prime proper $LI$-ideal and Boolean $LI$-ideal, (2) prime proper $LI$-ideal and $ILI$-ideal, (3) proper obstinate $LI$-ideal, (4) ultra $LI$-ideal.
LA - eng
KW - lattice implication algebra; $MTL$-algebra; (prime; ultra; obstinate; Boolean) $LI$-ideal; $ILI$-ideal; MTL-algebra; ILI-ideal
UR - http://eudml.org/doc/31149
ER -

References

top
  1. 10.1023/A:1022935811257, Czechoslovak Math. J. 53(128) (2003), 161–171. (2003) MR1962006DOI10.1023/A:1022935811257
  2. 10.1023/A:1022407325810, Czechoslovak Math. J. 48(123) (1998), 21–29. (1998) MR1614060DOI10.1023/A:1022407325810
  3. Monoidal t -norm based logic: Towards a logic for left-continuous t -norms, Fuzzy Sets Syst. 124 (2001), 271–288. (2001) MR1860848
  4. Metamathematics of Fuzzy Logic, Kluwer Academic Publishers, 1998. (1998) MR1900263
  5. 10.1023/A:1015122331293, Studia Logica 70 (2002), 183–192. (2002) MR1894392DOI10.1023/A:1015122331293
  6. On L I -ideals and prime L I -ideals of lattice implication alebras, J.  Korean Math. Soc. 36 (1999), 369–380. (1999) MR1690028
  7. L I -ideals in lattice implication algebras, Bull. Korean Math. Soc. 35 (1998), 13–24. (1998) MR1609010
  8. Fuzzy L I -ideals in lattice implication algebras, J.  Fuzzy Math. 7 (1999), 997–1003. (1999) Zbl0972.03550MR1734015
  9. 10.1016/S0020-0255(03)00159-2, Information Sciences 155 (2003), 157–175. (2003) MR2007038DOI10.1016/S0020-0255(03)00159-2
  10. 10.1093/jigpal/jzi034, Log.  J.  IGPL 13 (2005), 443–466. (2005) MR2163142DOI10.1093/jigpal/jzi034
  11. 10.1023/A:1021759209277, Czechoslovak Math.  J. 52(127) (2002), 463–468. (2002) MR1923253DOI10.1023/A:1021759209277
  12. 10.1007/s001530100088, Arch. Math. Logic 40 (2001), 467–473. (2001) Zbl1030.03048MR1854896DOI10.1007/s001530100088
  13. Non-classical Mathematical Logic and Approximate Reasoning, Science Press, Beijing, 2000. (Chinese) (2000) 
  14. M V -algebras, B L -algebras, R 0 -algebras and multiple-valued logic, Fuzzy Systems and Mathematics 16 (2002), 1–15. (Chinese) (2002) MR1911031
  15. Lattice implication algebras, J.  South West Jiaotong University 1 (1993), 20–27. (1993) Zbl0966.03524
  16. On filters of lattice implication algebras, J.  Fuzzy Math. 1 (1993), 251–260. (1993) MR1230317
  17. Lattice-valued Logic. An alternative approach to treat fuzziness and incomparability, Studies in Fuzzines and Soft Computing  132, Springer-Verlag, , 2003. (2003) MR2027329
  18. On fuzzy logic algebraic system  M T L , Advances in Systems and Applications 5 (2005), 475–483. (2005) 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.