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Dual Space of a Lattice as the Completion of a Pervin Space

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Relational and Algebraic Methods in Computer Science (RAMICS 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10226))

Abstract

We assume the reader is familiar with basic topology on the one hand and finite automata theory on the other hand. No proofs are given in this extended abstract.

J. Pin—Funded by the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme (grant agreement No. 670624) and by the DeLTA project (ANR-16-CE40-0007).

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Notes

  1. 1.

    Recall that \(u^{-1}L = \{x \in A^* \mid ux \in L\}\) and \(Lu^{-1} = \{x \in A^* \mid xu \in L\}\).

  2. 2.

    Formally, an epimorphism, but it is easy to see that in the category \(\mathbf {Pervin}\) epimorphisms coincide with surjective morphisms.

  3. 3.

    Let us define the characteristic function of an ultrafilter \(\mathcal {U}\) as the map from \(\mathcal {P}(A^*)\) to \(\{0, 1\}\) taking value 1 on \(\mathcal {U}\) and 0 elsewhere. It is easy to see that it is a valuation on \(\mathcal {P}(A^*)\). Conversely, if v is a valuation on \(\mathcal {P}(A^*)\), then \(v^{-1}(1)\) is an ultrafilter.

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Acknowledgements

I would like to thank Mai Gehrke and Serge Grigorieff for many fruitful discussions on Pervin spaces. I would also like to thank Daniela Petrişan for her critical help on categorical notions used in this paper. Encouragements from Hans-Peter A. Künzi and Marcel Erné were greatly appreciated.

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Authors and Affiliations

  1. IRIF, University Paris-Diderot and CNRS, Paris, France

    Jean-Éric Pin

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  1. Jean-Éric Pin
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Correspondence to Jean-Éric Pin .

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  1. Data61, CSIRO, Sydney, New South Wales, Australia

    Peter Höfner

  2. CNRS, Lyon, France

    Damien Pous

  3. University of Sheffield, Sheffield, United Kingdom

    Georg Struth

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Pin, JÉ. (2017). Dual Space of a Lattice as the Completion of a Pervin Space. In: Höfner, P., Pous, D., Struth, G. (eds) Relational and Algebraic Methods in Computer Science. RAMICS 2017. Lecture Notes in Computer Science(), vol 10226. Springer, Cham. https://doi.org/10.1007/978-3-319-57418-9_2

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  • DOI: https://doi.org/10.1007/978-3-319-57418-9_2

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