Non-commutative Quantales for Many-Valuedness in Applications

  • Patrik EklundEmail author
  • Ulrich Höhle
  • Jari Kortelainen
Conference paper
Part of the Communications in Computer and Information Science book series (CCIS, volume 610)


In this paper we show how the diversity of properties for quantales is well suited for describing multivalence in many-valued logic. Tensor products of quantales will play an important role in showing how more simple valuation scales can be tensored together to provide more complex valuation scales. In health care applications, this is typically seen for disorders and functioning. Classification of disorder is typically quite bivalent, whereas scales used in functioning classifications are multivalent. The role ‘not specified’ or ‘missing’ is shown to be of importance.


Assessment Logic Quantale 


  1. 1.
    Boole, G.: The Laws of Thought. Walton and Maberly, London (1853)zbMATHGoogle Scholar
  2. 2.
    Cunningham, J.V.: Logic and lyric. Mod. Philol. 51, 33–41 (1953)CrossRefGoogle Scholar
  3. 3.
    Diniz, M.A., Butters, S.M., Albert, M.A.D., Dew, M.A., Reynolds, C.F.: Late-life depression and risk of vascular dementia and Alzheimers disease: systematic review and meta-analysis of community-based cohort studies. Brit. J. Psychiatry 202, 329–335 (2013)CrossRefGoogle Scholar
  4. 4.
    Dougados, M., et al.: Prevalence of comorbidities in rheumatoid arthritis, evaluation of their monitoring: results of an international, cross-sectional study (COMORA). Ann. Rheum. Dis. 73, 62 (2013)CrossRefGoogle Scholar
  5. 5.
    Eklund, P.: Lative Logic Modelling Languages and Notation, in preparationGoogle Scholar
  6. 6.
    Eklund, P.: Lative logic accomodating the WHO family of international classifications. In: Cruz-Cunha, M.M., Miranda, I. (eds.) Encyclopedia of E-Health and Telemedicine. IGI Global (in print)Google Scholar
  7. 7.
    Eklund, H.-T., Eklund, P.: Logic and process drama. In: Thorkelsdóttir, R.B., Ragnardóttir Fliss, Á.H., Háskólaprent, Reykjavik, (eds.) Drama Boreale, Earth-Air-Water-Fire, pp. 103–115 (2013)Google Scholar
  8. 8.
    Eklund, P., Galán, M.A., Helgesson, R., Kortelainen, J.: Fuzzy terms. Fuzzy Sets Syst. 256, 211–235 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Eklund, P., Höhle, U., Kortelainen, J.: A survey on the categorical term construction with applications. Fuzzy Sets Syst. (2015)Google Scholar
  10. 10.
    Eklund, P., Galán, M.Á., Helgesson, R., Kortelainen, J., Moreno, G., Vázquez, C.: Towards categorical fuzzy logic programming. In: Masulli, F. (ed.) WILF 2013. LNCS, vol. 8256, pp. 109–121. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  11. 11.
    Gutiérrez García, J., Höhle, U., Kubiak, T.: Tensor products of complete lattices and their application in constructing quantales, preprintGoogle Scholar
  12. 12.
    Höhle, U.: Commutative, residuated \(\ell \)-monoids. In: Höhle, U., Klement, E.P. (eds.) Non-classical Logics and their Applications to Fuzzy Subsets, pp. 53–106. Kluwer Academic Publishers, Dordrecht (1995)CrossRefGoogle Scholar
  13. 13.
    Höhle, U.: Topological representation of right-sided and idempotent quantales. Semigroup Forum 90, 648–659 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Höhle, U.: Prime elements of non-integral quantales and their applications. Order 32, 329–334 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Lyketsos, C.G., Steele, C., Baker, L., Galik, E., Kopunek, S., Steinberg, M., Warren, A.: Major and minor depression in Alzheimer’s disease: prevalence and impact. J. Neuropsychiatry Clin. Neurosci. 9, 556–561 (1997)CrossRefGoogle Scholar
  16. 16.
    Mulvey, C.J., Pelletier, J.W.: On the quantisation of the calculus of relations. In: CMS Proceedings, vol. 13, pp. 345–360. American Mathematical Society, Providence RI (1992)Google Scholar
  17. 17.
    Pelletier, J.W., Rosický, J.: Simple involutive quantales. J. Algebra 195, 367–386 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Thorpe, L.: Depression vs. Dementia: how do we assess? The Canadian Review of Alzheimers Disease and Other Dementias, pp. 17–21, September 2009Google Scholar
  19. 19.
    Wilson, R.S., Capuano, A.W., Boyle, P.A., Hoganson, G.M., Hizel, L.P., Shah, R.C., Nag, S., Schneider, J.A., Arnold, S.E., Bennett, D.A.: Clinical-pathologic study of depressive symptoms and cognitive decline in old age. Neurology 83, 702–709 (2014)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Patrik Eklund
    • 1
    Email author
  • Ulrich Höhle
    • 2
  • Jari Kortelainen
    • 3
  1. 1.Department of Computing ScienceUmeå UniversityUmeåSweden
  2. 2.FB C Bergische Universität WuppertalWuppertalGermany
  3. 3.Department of Electrical Engineering and Information TechnologyMikkeli University of Applied SciencesMikkeliFinland

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