License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.CALCO.2017.18
URN: urn:nbn:de:0030-drops-80314
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2017/8031/
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Milius, Stefan

Proper Functors and their Rational Fixed Point

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LIPIcs-CALCO-2017-18.pdf (0.5 MB)

Abstract

The rational fixed point of a set functor is well-known to capture the behaviour of finite coalgebras. In this paper we consider functors on algebraic categories. For them the rational fixed point may no longer be a subcoalgebra of the final coalgebra. Inspired by Ésik and Maletti's notion of proper semiring, we introduce the notion of a proper functor. We show that for proper functors the rational fixed point is determined as the colimit of all coalgebras with a free finitely generated algebra as carrier and it is a subcoalgebra of the final coalgebra. Moreover, we prove that a functor is proper if and only if that colimit is a subcoalgebra of the final coalgebra. These results serve as technical tools for soundness and completeness proofs for coalgebraic regular expression calculi, e.g. for weighted automata.

BibTeX - Entry

@InProceedings{milius:LIPIcs:2017:8031,
  author =	{Stefan Milius},
  title =	{{Proper Functors and their Rational Fixed Point}},
  booktitle =	{7th Conference on Algebra and Coalgebra in Computer Science (CALCO 2017)},
  pages =	{18:1--18:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-033-0},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{72},
  editor =	{Filippo Bonchi and Barbara K{\"o}nig},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2017/8031},
  URN =		{urn:nbn:de:0030-drops-80314},
  doi =		{10.4230/LIPIcs.CALCO.2017.18},
  annote =	{Keywords: proper functor, proper semiring, coalgebra, rational fixed point}
}

Keywords: proper functor, proper semiring, coalgebra, rational fixed point
Collection: 7th Conference on Algebra and Coalgebra in Computer Science (CALCO 2017)
Issue Date: 2017
Date of publication: 17.11.2017

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