Download PDF - Interior and closure operators on bounded commutative residuated l-monoidsAll rights reserved. Copyright exceptions and limitations apply

ArticleOriginal scientific text

Title

Interior and closure operators on bounded commutative residuated l-monoids

Authors 1, 1

Affiliations

  1. Department of Algebra and Geometry, Faculty of Sciences, Palacký University, Tomkova 40, CZ-779 00 Olomouc, Czech Republic

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.

Keywords

ENG
residuated l-monoid, residuated lattice, closure operator, BL-algebra, MV-algebra

Bibliography

  1. [1] P. Bahls, J. Cole, N. Galatos, P. Jipsen and C. Tsinakis, Cancellative residuated lattices, Alg. Univ. 50 (2003), 83-106.
  2. [2] K. Blount and C. Tsinakis, The structure of residuated lattices, Intern. J. Alg. Comp. 13 (2003), 437-461.
  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.
  4. [4]A. Dvurečenskij and S. Pulmannová, New Trends in Quantum Structures, Kluwer Acad. Publ., Dordrecht-Boston-London 2000.
  5. [5] A. Dvurečenskij and J. Rachůnek, Bounded commutative residuated l-monoids with general comparability and states, Soft Comput. 10 (2006), 212-218.
  6. [6] P. Hájek, Metamathematics of Fuzzy Logic, Kluwer, Amsterdam 1998.
  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.
  8. [8] J. Kühr, Dually Residuated Lattice Ordered Monoids, Ph. D. Thesis, Palacký Univ., Olomouc 2003.
  9. [9] J. Rachůnek, DRl-semigroups and MV-algebras, Czechoslovak Math. J. 48 (1998), 365-372.
  10. [10] J. Rachůnek, MV-algebras are categorically equivalent to a class of DRl1(i)-semigroups, Math. Bohemica 123 (1998), 437-441.
  11. [11] J. Rachůnek, A duality between algebras of basic logic and bounded representable DRl-monoids, Math. Bohemica 126 (2001), 561-569.
  12. [12] J. Rachůnek and V. Slezák, Negation in bounded commutative DRl-monoids, Czechoslovak Math. J. 56 (2006), 755-763.
  13. [13] J. Rachůnek and D. Šalounová, Local bounded commutative residuated l-monoids, Czechoslovak Math. J. 57 (2007), 395-406.
  14. [14] J. Rachůnek and F. Švrček, MV-algebras with additive closure operators, Acta Univ. Palacký, Mathematica 39 (2000), 183-189.
  15. [15] H. Rasiowa and R. Sikorski, The Mathematics of Metamathematics, Panstw. Wyd. Nauk., Warszawa 1963.
  16. [16] K.L.N. Swamy, Dually residuated lattice ordered semigroups, Math. Ann. 159 (1965), 105-114.
  17. [17] E. Turunen, Mathematics Behind Fuzzy Logic, Physica-Verlag, Heidelberg-New York 1999.
Discussiones Mathematicae - General Algebra and Applications
2008/28/1
Export to PBN, Opens in new tab
Pages:
11-27
Main language of publication
English
Received
2005-04-20
Accepted
2007-07-04
Published
2008

DOI: 10.7151/dmgaa.1132

Exact and natural sciences