Abstract
A function on an algebra is congruence preserving if, for any congruence, it maps pairs of congruent elements onto pairs of congruent elements. We show that on the algebra of full binary trees whose leaves are labeled by letters of an alphabet containing at least three letters, a function is congruence preserving if and only if it is polynomial. This exhibits an example of a non commutative and non associative 1–affine complete algebra. As far as we know, it is the first example of such an algebra.
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Acknowledgements
We thank the anonymous referees for their extremely careful reading and comments which helped in shortening the proof of the main theorem and improving the presentation of the paper.
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Presented by M. Haviar.
To the memory of Kate Karagueuzian-Gibbons and Giliane Arnold.
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Arnold, A., Cégielski, P., Grigorieff, S. et al. Affine completeness of the algebra of full binary trees. Algebra Univers. 81, 55 (2020). https://doi.org/10.1007/s00012-020-00690-6
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DOI: https://doi.org/10.1007/s00012-020-00690-6