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Quotients of d-Frames

  • Tomáš JaklEmail author
  • Achim Jung
  • Aleš Pultr
Article
  • 27 Downloads

Abstract

It is shown that every d-frame admits a complete lattice of quotients. Quotienting may be triggered by a binary relation on one of the two constituent frames, or by changes to the consistency or totality structure, but as these are linked by the reasonableness conditions of d-frames, the result in general will be that both frames are factored and both consistency and totality are increased.

Keywords

Quotient d-Frame (Quasi)-congruence Factorization system 

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Notes

Acknowledgements

The results reported in this paper were obtained in Spring 2017, during visits by the third author to Birmingham and the first two authors to Prague. We gratefully acknowledge the hospitality and financial support extended to us by our two universities. The work was supported by the Grant SVV–2017–260452 and by the CE-ITI Grant, GAČR P202/12/G061. We are also grateful to the anonymous referee for spotting several hidden typos.

References

  1. 1.
    Arieli, O., Avron, A.: Reasoning with logical bilattices. J. Logic Lang. Inf. 5, 25–63 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Abramsky, S.: Semantics of programming languages. Course Notes (1991)Google Scholar
  3. 3.
    Adámek, J., Herrlich, H., Strecker, G.: Abstract and Concrete Categories. Wiley, New York (1990)zbMATHGoogle Scholar
  4. 4.
    Abramsky, S., Jung, A.: Domain theory. In: Abramsky, S., Gabbay, D.M., Maibaum, T.S.E. (eds.) Semantic Structures, Volume 3 of Handbook of Logic in Computer Science, pp. 1–168. Clarendon Press, Oxford (1994)Google Scholar
  5. 5.
    Belnap, N.D.: How a computer should think. In: Ryle, G. (ed.) Contemporary Aspects of Philosophy, pp. 30–56. Oriel Press, Newcastle upon Tyne (1976)Google Scholar
  6. 6.
    Belnap, N.D.: A useful four-valued logic. In: Dunn, J.M., Epstein, G. (eds.) Modern Uses of Multiple-Valued Logic, pp. 8–37. Reidel Publishing Company, Dordrecht (1977)Google Scholar
  7. 7.
    Carollo, I.M., Moshier, M.A.: Extremal Epimorphisms in \({\sf dFrm}\)’ and an Isbell-Type Density Theorem. Chapman University, Orange (2017)Google Scholar
  8. 8.
    Gierz, G., Hofmann, K.H., Keimel, K., Lawson, J.D., Mislove, M., Scott, D.S.: Continuous Lattices and Domains, Volume 93 of Encyclopedia of Mathematics and Its Applications. Cambridge University Press, Cambridge (2003)zbMATHGoogle Scholar
  9. 9.
    Jakl, T., Jung, A.: Free constructions and coproducts of d-frames. In: CALCO 2017, Leibniz International Proceedings in Informatics. Dagstuhl Publishing (2017)Google Scholar
  10. 10.
    Jung, A., Moshier, M.A.: On the bitopological nature of Stone duality. Technical Report CSR-06-13, School of Computer Science, The University of Birmingham (2006)Google Scholar
  11. 11.
    Johnstone, P.T.: Stone Spaces, Volume 3 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (1982)Google Scholar
  12. 12.
    Mac Lane, S.: Categories for the Working Mathematician, Volume 5 of Graduate Texts in Mathematics. Springer, Berlin (1971)CrossRefzbMATHGoogle Scholar
  13. 13.
    Picado, J., Pultr, A.: Frames and Locales: Topology Without Points. Frontiers in Mathematics. Birkhäuser, Basel (2012)CrossRefzbMATHGoogle Scholar
  14. 14.
    Rivieccio, U.: An Algebraic Study of Bilattice-Based Logics. PhD thesis, University of Barcelona (2010)Google Scholar
  15. 15.
    Scott, D.S.: Domains for denotational semantics. In: Nielson, M., Schmidt, E.M. (eds.) International Colloquium on Automata, Languages and Programs, Volume 140 of Lecture Notes in Computer Science, pp. 577–613. Springer (1982)Google Scholar
  16. 16.
    Smyth, M.B.: Topology. In: Abramsky, S., Gabbay, D.M., Maibaum, T.S.E. (eds.) Handbook of Logic in Computer Science, vol. 1, pp. 641–761. Clarendon Press, Oxford (1992)Google Scholar
  17. 17.
    Vickers, S.J.: Topology via Logic, Volume 5 of Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, Cambridge (1989)Google Scholar

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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of Applied MathematicsCharles UniversityPragueCzech Republic
  2. 2.School of Computer ScienceThe University of BirminghamBirminghamUK

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