Skip to main content
SpringerLink
Log in

The p-adic Integers as Final Coalgebra

  • Conference paper
  • First Online:
Logic, Language, Information, and Computation (WoLLIC 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9160))

  • 386 Accesses

Abstract

We express the classical p-adic integers \(\hat{\mathbb {Z}}_p\), as a metric space, as the final coalgebra to a certain endofunctor. We realize the addition and the multiplication on \(\hat{\mathbb {Z}}_p\) as the coalgebra maps from \(\hat{\mathbb {Z}}_p\times \hat{\mathbb {Z}}_p\).

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 34.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 44.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Adámek, J.: Free algebras and automata realizations in the language of categories. Comment. Math. Univ. Carolinae 14, 589–602 (1974)

    Google Scholar 

  2. Adámek, J. Milius, S., Moss, L.S.: Initial Algebra and Terminal Coalgebra: A survey (work in progress)

    Google Scholar 

  3. Barr, M.: Terminal coalgebra in well founded set theory. Theret. Comput. Sci. 144, 299–314 (1993)

    Article  MathSciNet  Google Scholar 

  4. Cuoco, A.A.: Visualizing the \(p\)-adic integers. Amer. Math. Monthly 98(4), 355–364 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  5. Leinster, L.: A general theory of self-similar spaces. Adv. Math. 226(4), 2935–3017 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  6. Rutten, J.J.M.M.: Universal coalgebra: a theory of systems. Modern algebra and its applications. Theoret. Comput. Sci. 249(1), 3–80 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  7. Serre, J.P.: A Course in Arithmetic. Graduate Texts in Mathematics, vol. 7, viii+115 p. Springer, New York (1973)

    Google Scholar 

Download references

Acknowledgement

I would like to thank Larry Moss introducing me to this subject and assisting me at various stages of this project, Robert Rose for various discussions about this project and finally Michael Mandell for being very supportive as an adviser and allowing me the freedom to explore different areas of Mathematics.

Author information

Authors and Affiliations

  1. Department of Mathematics, Indiana University, Bloomington, 47405, USA

    Prasit Bhattacharya

Authors
  1. Prasit Bhattacharya
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Prasit Bhattacharya .

Editor information

Editors and Affiliations

  1. Nuance Communications, Sunnyvale, California, USA

    Valeria de Paiva

  2. Universidade Federal de Pernambuco, Centro de Informática, Recife, Pernambuco, Brazil

    Ruy de Queiroz

  3. Department of Mathematics, Indiana University Bloomington, Bloomington, Indiana, USA

    Lawrence S. Moss

  4. nDepartment of Computer Science, Indiana University Bloomington, Bloomington, Indiana, USA

    Daniel Leivant

  5. Universidade Federal de Pernambuco, Centro de Informática, Recife, Pernambuco, Brazil

    Anjolina G. de Oliveira

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Bhattacharya, P. (2015). The p-adic Integers as Final Coalgebra . In: de Paiva, V., de Queiroz, R., Moss, L., Leivant, D., de Oliveira, A. (eds) Logic, Language, Information, and Computation. WoLLIC 2015. Lecture Notes in Computer Science(), vol 9160. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-47709-0_14

Download citation

  • DOI: https://doi.org/10.1007/978-3-662-47709-0_14

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-47708-3

  • Online ISBN: 978-3-662-47709-0

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation