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1 results

  • 10 - Fixed Points Determined by Finite Behaviour

  • Jiří Adámek, Czech Technical University in Prague, Stefan Milius, Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany, Lawrence S. Moss, Indiana University, Bloomington
  • Book:
    Initial Algebras and Terminal Coalgebras
    Published online:
    30 January 2025
    Print publication:
    06 February 2025, pp 319-362

  • Summary

    The rational fixed point of an endofunctor is a fixed point which is in general different from its initial algebra and its terminal coalgebra. It collects precisely the behaviours of all ‘finite’ coalgebras of a given endofunctor. For sets, they are those with finitely many states. Examples of rational fixed points include regular languages, eventually periodic and rational streams, etc. To study the rational fixed point in categories beyond sets, we discuss locally finitely presentable categories, and we do so in some detail. We characterize the rational fixed point as an initial iterative algebra. The chapter goes into details on many examples, such as rational fixed points in nominal sets. It discusses the rational fixed point and several other fixed points as well, and it summarizes much of what is known about them.