Abstract
Pavlović and Pratt obtained several presentations of sets such as the half-open interval [0, 1) as the final coalgebra of a very basic functor on the category of sets, namely product with the natural numbers. We re-prove and extend some of their results, and we establish some new presentations as well. More importantly in this paper, we exhibit several corecursive algebra structures on sets of real numbers, and we connect to continued fractions and to linear fractional transformations. We present a general result which, under hypotheses, shows that a corecursive algebra has as a subalgebra the final coalgebra with the inverse structure.
Supported by grant #586136 from the Simons Foundation.
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Acknowledgments
We are grateful to Jiří Adámek and Stefan Milius for discussions of many matters related to this paper. We thank Helle Hansen for pointing out serious errors in a previous version, and for her encouragement. We also thank the referees for their criticism of an earlier version of the paper, and for their suggestions on how to improve it. All errors and shortcomings of this paper are our own.
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Moss, L.S., Noquez, V. (2022). Corecursive Algebras in Nature. In: Hansen, H.H., Zanasi, F. (eds) Coalgebraic Methods in Computer Science. CMCS 2022. Lecture Notes in Computer Science, vol 13225. Springer, Cham. https://doi.org/10.1007/978-3-031-10736-8_8
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DOI: https://doi.org/10.1007/978-3-031-10736-8_8
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