Interior and closure operators on bounded commutative residuated l-monoids

Jiří Rachůnek; Filip Švrček

Discussiones Mathematicae - General Algebra and Applications (2008)

  • Volume: 28, Issue: 1, page 11-27
  • ISSN: 1509-9415

Abstract

top
Topological Boolean algebras are generalizations of topological spaces defined by means of topological closure and interior operators, respectively. The authors in [14] generalized topological Boolean algebras to closure and interior operators of MV-algebras which are an algebraic counterpart of the Łukasiewicz infinite valued logic. In the paper, these kinds of operators are extended (and investigated) to the wide class of bounded commutative Rl-monoids that contains e.g. the classes of BL-algebras (i.e., algebras of the Hájek's basic fuzzy logic) and Heyting algebras as proper subclasses.

How to cite

top

Jiří Rachůnek, and Filip Švrček. "Interior and closure operators on bounded commutative residuated l-monoids." Discussiones Mathematicae - General Algebra and Applications 28.1 (2008): 11-27. <http://eudml.org/doc/276908>.

@article{JiříRachůnek2008,
abstract = {Topological Boolean algebras are generalizations of topological spaces defined by means of topological closure and interior operators, respectively. The authors in [14] generalized topological Boolean algebras to closure and interior operators of MV-algebras which are an algebraic counterpart of the Łukasiewicz infinite valued logic. In the paper, these kinds of operators are extended (and investigated) to the wide class of bounded commutative Rl-monoids that contains e.g. the classes of BL-algebras (i.e., algebras of the Hájek's basic fuzzy logic) and Heyting algebras as proper subclasses.},
author = {Jiří Rachůnek, Filip Švrček},
journal = {Discussiones Mathematicae - General Algebra and Applications},
keywords = {residuated l-monoid; residuated lattice; closure operator; BL-algebra; MV-algebra},
language = {eng},
number = {1},
pages = {11-27},
title = {Interior and closure operators on bounded commutative residuated l-monoids},
url = {http://eudml.org/doc/276908},
volume = {28},
year = {2008},
}

TY - JOUR
AU - Jiří Rachůnek
AU - Filip Švrček
TI - Interior and closure operators on bounded commutative residuated l-monoids
JO - Discussiones Mathematicae - General Algebra and Applications
PY - 2008
VL - 28
IS - 1
SP - 11
EP - 27
AB - Topological Boolean algebras are generalizations of topological spaces defined by means of topological closure and interior operators, respectively. The authors in [14] generalized topological Boolean algebras to closure and interior operators of MV-algebras which are an algebraic counterpart of the Łukasiewicz infinite valued logic. In the paper, these kinds of operators are extended (and investigated) to the wide class of bounded commutative Rl-monoids that contains e.g. the classes of BL-algebras (i.e., algebras of the Hájek's basic fuzzy logic) and Heyting algebras as proper subclasses.
LA - eng
KW - residuated l-monoid; residuated lattice; closure operator; BL-algebra; MV-algebra
UR - http://eudml.org/doc/276908
ER -

References

top
  1. [1] P. Bahls, J. Cole, N. Galatos, P. Jipsen and C. Tsinakis, Cancellative residuated lattices, Alg. Univ. 50 (2003), 83-106. Zbl1092.06012
  2. [2] K. Blount and C. Tsinakis, The structure of residuated lattices, Intern. J. Alg. Comp. 13 (2003), 437-461. Zbl1048.06010
  3. [3] R.O.L. Cignoli, I.M.L. D'Ottaviano and D. Mundici, Algebraic Foundations of Many-valued Reasoning, Kluwer Acad. Publ., Dordrecht-Boston-London 2000. Zbl0937.06009
  4. [4]A. Dvurečenskij and S. Pulmannová, New Trends in Quantum Structures, Kluwer Acad. Publ., Dordrecht-Boston-London 2000. Zbl0987.81005
  5. [5] A. Dvurečenskij and J. Rachůnek, Bounded commutative residuated l-monoids with general comparability and states, Soft Comput. 10 (2006), 212-218. Zbl1096.06008
  6. [6] P. Hájek, Metamathematics of Fuzzy Logic, Kluwer, Amsterdam 1998. Zbl0937.03030
  7. [7] P. Jipsen and C. Tsinakis, A survey of residuated lattices, Ordered algebraic structures (ed. J. Martinez), Kluwer Acad. Publ. Dordrecht (2002), 19-56. Zbl1070.06005
  8. [8] J. Kühr, Dually Residuated Lattice Ordered Monoids, Ph. D. Thesis, Palacký Univ., Olomouc 2003. Zbl1066.06008
  9. [9] J. Rachůnek, DRl-semigroups and MV-algebras, Czechoslovak Math. J. 48 (1998), 365-372. Zbl0952.06014
  10. [10] J. Rachůnek, MV-algebras are categorically equivalent to a class of D R l 1 ( i ) -semigroups, Math. Bohemica 123 (1998), 437-441. Zbl0934.06014
  11. [11] J. Rachůnek, A duality between algebras of basic logic and bounded representable DRl-monoids, Math. Bohemica 126 (2001), 561-569. Zbl0979.03049
  12. [12] J. Rachůnek and V. Slezák, Negation in bounded commutative DRl-monoids, Czechoslovak Math. J. 56 (2006), 755-763. Zbl1164.06325
  13. [13] J. Rachůnek and D. Šalounová, Local bounded commutative residuated l-monoids, Czechoslovak Math. J. 57 (2007), 395-406. Zbl1174.06331
  14. [14] J. Rachůnek and F. Švrček, MV-algebras with additive closure operators, Acta Univ. Palacký, Mathematica 39 (2000), 183-189. Zbl1039.06005
  15. [15] H. Rasiowa and R. Sikorski, The Mathematics of Metamathematics, Panstw. Wyd. Nauk., Warszawa 1963. Zbl0122.24311
  16. [16] K.L.N. Swamy, Dually residuated lattice ordered semigroups, Math. Ann. 159 (1965), 105-114. Zbl0135.04203
  17. [17] E. Turunen, Mathematics Behind Fuzzy Logic, Physica-Verlag, Heidelberg-New York 1999. Zbl0940.03029

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.