Soft Computing

, Volume 21, Issue 10, pp 2561–2574 | Cite as

Quantale algebras as lattice-valued quantales



In this paper, we present an investigation of quantale algebras (Q-algebras for short) as lattice-valued quantales (Q-quantales for short). First, we prove that the set of all fuzzy ideals of a commutative ring with appropriate operations is a [0, 1]-quantale. Furthermore, we discuss some properties of localic nuclei on Q-algebras, and show that the category of Q-algebras with the quantale structures being frames is a full reflective subcategory of the category of Q-algebras. From this result, we can conclude that the category of L-frames is a full reflective subcategory of the category of L-quantales, where L is a frame. Finally, we build and characterize the Q-quantale completions of a Q-ordered semigroup.


Quantale Quantale algebra Q-ordered set Lattice-valued quantale 



This work is supported by the National Natural Science Foundation of China (Grant Nos. 11301316, 11531009), the Natural Science Program for Basic Research of Shaanxi Province (Grant No. 2015JM1020) and the Fundamental Research Funds for the Central Universities (Grant Nos. GK201302003, GK201501001). The authors would like to thank the referees and the editors for their valuable comments and suggestions.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. Abramsky S, Vickers S (1993) Quantales, observational logic and process semantics. Math Struct Comput Sci 3:161–227MathSciNetCrossRefMATHGoogle Scholar
  2. Adámek J, Herrlich H, Strecker GE (1990) Abstract and Concrete Categories: The Joy of Cats. Wiley, New YorkMATHGoogle Scholar
  3. Bělohlávek R (1999) Fuzzy Galois connections. Math Log Q 45:497–504MathSciNetCrossRefMATHGoogle Scholar
  4. Bělohlávek R (2001) Fuzzy closure operators. J Math Anal Appl 262:473–489MathSciNetCrossRefMATHGoogle Scholar
  5. Bělohlávek R (2002) Fuzzy Relational Systems: Foundations and Principles. Kluwer Academic Publishers/Plenum Publishers, New YorkCrossRefMATHGoogle Scholar
  6. Bělohlávek R (2004) Concept lattices and order in fuzzy logic. Ann Pure Appl Log 128:277–298MathSciNetCrossRefMATHGoogle Scholar
  7. Bělohlávek R, Vychodil V (2006) Algebras with fuzzy equalities. Fuzzy Sets Syst 157:161–201MathSciNetCrossRefMATHGoogle Scholar
  8. Borceux F, Rosický J, van den Bossche G (1989) Quantales and \(C^*\)-algebras. J Lond Math Soc 40:398–404MathSciNetCrossRefMATHGoogle Scholar
  9. Coniglio ME, Miraglia F (2000) Non-commutative topology and quantales. Stud Log 65:223–236MathSciNetCrossRefMATHGoogle Scholar
  10. Davey BA, Priestley HA (2002) Introduction to Lattices and Order, 2nd edn. Cambridge University Press, CambridgeCrossRefMATHGoogle Scholar
  11. Dilworth RP (1939) Non-commutative residuated lattices. Trans Am Math Soc 46:426–444MathSciNetCrossRefMATHGoogle Scholar
  12. Fan L (2001) A new approach to quantitative domain theory. Electron Notes Theor Comput Sci 45:77–87CrossRefMATHGoogle Scholar
  13. Galatos N, Jipsen P, Kowalski T, Ono H (2007) Residuated Lattices: An Algebraic Glimpse at Substructural Logics. In: Studies in Logics and th Foundations of Mathematics, vol 151. Elsevier, AmsterdamGoogle Scholar
  14. Girard JY (1997) Linear logic. Theor Comput Sci 50:1–102MathSciNetCrossRefMATHGoogle Scholar
  15. Goguen JA (1967) \(L\)-fuzzy sets. J Math Anal Appl 18:145–174MathSciNetCrossRefMATHGoogle Scholar
  16. Han SW, Zhao B (2008) The quantale completion of ordered semigroup. Acta Math Sin Chin Ser 51(6):1081–1088MathSciNetMATHGoogle Scholar
  17. Hungerford TW (2003) Algebra. Springer, New YorkMATHGoogle Scholar
  18. Johnstone PT (1982) Stone Spaces. Cambridge University Press, CambridgeMATHGoogle Scholar
  19. Joyal A, Tierney M (1984) An extension of the Galois theory of Grothendieck. Mem Am Math Soc 309:1–71MathSciNetMATHGoogle Scholar
  20. Kehayopulu N, Tsingelis M (2002) Fuzzy sets in ordered groupoids. Semigroup Forum 65:128–132MathSciNetCrossRefMATHGoogle Scholar
  21. Kehayopulu N, Tsingelis M (2005) Fuzzy bi-ideals in ordered semigroups. Inf Sci 171:13–28MathSciNetCrossRefMATHGoogle Scholar
  22. Kelly GM (1982) Basic Concepts of Enriched Category Theory. London Mathematical Society Lecture Notes Series, vol 64. Cambridge University Press, CambridgeGoogle Scholar
  23. Kruml D, Paseka J (2008) Algebraic and categorical aspects of quantales. Handb Algebra 5:323–362MathSciNetCrossRefMATHGoogle Scholar
  24. Lai H, Zhang D (2006) Many-valued complete distributivity. arXiv:math/0603590v2
  25. Li YM, Li ZH (1999) Quantales and process semantics of bisimulation. Acta Math Sin Chin Ser 42:313–320MathSciNetMATHGoogle Scholar
  26. Li YM, Zhou M, Li ZH (2002) Projective and injective objects in the category of quantales. J Pure Appl Algebra 176(2):249–258MathSciNetMATHGoogle Scholar
  27. Liu WJ (1982) Fuzzy invariant subgroups and fuzzy ideals. Fuzzy Sets Syst 8:133–139MathSciNetCrossRefMATHGoogle Scholar
  28. Liu WJ (1983) Operations on fuzzy ideals. Fuzzy Sets Syst 11:37–41MathSciNetGoogle Scholar
  29. Mulvey CJ (1986)&. Rend Circ Mat Palermo (2) Suppl 12(2):99–104Google Scholar
  30. Mulvey CJ, Pelletier JW (2001) On the quantisation of points. J Pure Appl Algebra 159:231–295MathSciNetCrossRefMATHGoogle Scholar
  31. Pan FF, Han SW (2014) Free \(Q\)-algebras. Fuzzy Sets Syst 247:138–150CrossRefMATHGoogle Scholar
  32. Paseka J (2000) A note on Girard bimodules. Int J Theor Phys 39:805–811MathSciNetCrossRefMATHGoogle Scholar
  33. Resende P (2001) Quantales, finite observations and strong bisimulation. Theor Comput Sci 254:95–149MathSciNetCrossRefMATHGoogle Scholar
  34. Rodabaugh SE (1999) Powerset operator foundations for poslat fuzzy set theories and topologies. In: Höhle U, Rodabaugh SE (eds) Mathematics of Fuzzy Sets: Logic, Topology, and Measure Theory. The Handbooks of Fuzzy Sets Series, vol 3. Kluwer Academic Publishers, Boston, pp 91–116 (Chapter 2)Google Scholar
  35. Rosenfeld A (1971) Fuzzy groups. J Math Anal Appl 35:512–517MathSciNetCrossRefMATHGoogle Scholar
  36. Rosenthal KI (1990) Quantales and their Applications. Longman Scientific & Technical, New YorkMATHGoogle Scholar
  37. Rosenthal KI (1994) Modules over a quantale and models for the operator ! in linear logic. Cah Topologie Géom Différ Catég 35:329–333MathSciNetMATHGoogle Scholar
  38. Russo C (2007) Quantale modules, with applications to logic and image processing. PhD thesis, Department of Mathematics and Computer Science, University of Salerno, SalernoGoogle Scholar
  39. Solovyov SA (2008a) On the category \(Q\)-Mod. Algebra Univers 58:35–58MathSciNetCrossRefMATHGoogle Scholar
  40. Solovyov SA (2008b) A representation theorem for quantale algebras. Contrib Gen Algebra 18:189–198MathSciNetMATHGoogle Scholar
  41. Solovyov SA (2010) From quantale algebroids to topological spaces: fixed- and variable-basis approaches. Fuzzy Sets Syst 161:1270–1287MathSciNetCrossRefMATHGoogle Scholar
  42. Solovyov SA (2011) A note on nuclei of quantale algebras. Bull Sect Log 40:91–112MathSciNetMATHGoogle Scholar
  43. Stubbe I (2006) Categorical structures enriched in a quantaloid: tensored and cotensored categories. Theory Appl Categ 16:283–306MathSciNetMATHGoogle Scholar
  44. Vickers S (1989) Topology via Logic. Cambridge University Press, CambridgeMATHGoogle Scholar
  45. Wagner KR (1994) Solving recursive domain equations with enriched categories. PhD Thesis, School of Computer Science, Carnegie Mellon University, PittsburghGoogle Scholar
  46. Wagner KR (1997) Liminf convergence in \(\Omega \)-categories. Theor Comput Sci 184:61–104MathSciNetCrossRefMATHGoogle Scholar
  47. Wang K (2012) Some researches on fuzzy domains and fuzzy quantales. PhD Thesis, Department of Mathematics, Shaanxi Normal University, Xi’an (in Chinese)Google Scholar
  48. Wang R, Zhao B (2010) Quantale algebra and its algebraic ideal. Fuzzy Syst Math 24:44–49 (in Chinese)Google Scholar
  49. Wang K, Zhao B (2012) Join-completions of \(L\)-ordered sets. Fuzzy Sets Syst 199:92–107MathSciNetCrossRefMATHGoogle Scholar
  50. Wang K, Zhao B (2013) Some properties of the category of fuzzy quantales. J Shaanxi Norm Univ (Nat Sci Ed) 41(3):1–6 (in Chinese)Google Scholar
  51. Wang K, Zhao B (2016a) On embeddings of \(Q\)-algebras. Houst J Math 42(1):73–90MathSciNetMATHGoogle Scholar
  52. Wang K, Zhao B (2016b) A representation theorem for Girard \(Q\)-algebras. J Mult Valued Log Soft Comput (in press)Google Scholar
  53. Ward M (1938) Structure residuation. Ann Math 39(3):558–569MathSciNetCrossRefMATHGoogle Scholar
  54. Ward M, Dilworth RP (1939) Residuated lattices. Trans Am Math Soc 45:335–354MathSciNetCrossRefMATHGoogle Scholar
  55. Xie XY, Tang J (2008) Fuzzy radicals and prime fuzzy ideals of ordered semigroups. Inf Sci 178:4357–4374MathSciNetCrossRefMATHGoogle Scholar
  56. Xie WX, Zhang QY, Fan L (2009) The Dedekind–MacNeille completions for fuzzy posets. Fuzzy Sets Syst 160:2292–2316MathSciNetCrossRefMATHGoogle Scholar
  57. Yao W, Lu LX (2009) Fuzzy Galois connections on fuzzy posets. Math Log Q 55:84–91MathSciNetCrossRefMATHGoogle Scholar
  58. Yao W (2010) Quantitative domains via fuzzy sets: part I: continuity of fuzzy directed complete posets. Fuzzy Sets Syst 161:973–987MathSciNetCrossRefMATHGoogle Scholar
  59. Yao W (2011) An approach to fuzzy frames via fuzzy posets. Fuzzy Sets Syst 166:75–89MathSciNetCrossRefMATHGoogle Scholar
  60. Yao W (2012) A survey of fuzzifications of frames, the Papert–Papert–Isbell adjunction and sobriety. Fuzzy Sets Syst 190:63–81MathSciNetCrossRefMATHGoogle Scholar
  61. Yetter D (1990) Quantales and (noncommutative) linear logic. J Symb Log 55:41–64MathSciNetCrossRefMATHGoogle Scholar
  62. Zadeh LA (1971) Similarity relations and fuzzy orderings. Inf Sci 3:177–200MathSciNetCrossRefMATHGoogle Scholar
  63. Zhang QY, Fan L (2005) Continuity in quantitative domains. Fuzzy Sets Syst 154:118–131MathSciNetCrossRefMATHGoogle Scholar
  64. Zhang QY, Xie WX, Fan L (2009) Fuzzy complete lattices. Fuzzy Sets Syst 160:2275–2291MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.College of Mathematics and Information ScienceShaanxi Normal UniversityXi’anPeople’s Republic of China

Personalised recommendations