Quotients of d-Frames

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.

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References

  1. 1.

    Arieli, O., Avron, A.: Reasoning with logical bilattices. J. Logic Lang. Inf. 5, 25–63 (1996)

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    Abramsky, S.: Semantics of programming languages. Course Notes (1991)

  3. 3.

    Adámek, J., Herrlich, H., Strecker, G.: Abstract and Concrete Categories. Wiley, New York (1990)

    Google 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)

    Google 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)

  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)

  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)

    Google Scholar 

  13. 13.

    Picado, J., Pultr, A.: Frames and Locales: Topology Without Points. Frontiers in Mathematics. Birkhäuser, Basel (2012)

    Google Scholar 

  14. 14.

    Rivieccio, U.: An Algebraic Study of Bilattice-Based Logics. PhD thesis, University of Barcelona (2010)

  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)

  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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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.

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Correspondence to Tomáš Jakl.

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Communicated by Jorge Picado.

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Jakl, T., Jung, A. & Pultr, A. Quotients of d-Frames. Appl Categor Struct 27, 261–275 (2019). https://doi.org/10.1007/s10485-018-09553-7

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Keywords

  • Quotient
  • d-Frame
  • (Quasi)-congruence
  • Factorization system