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PreprintarXiv licencearXiv

Constructive algebraic methods for bisymmetric means

  • Pál Burai
  • Gergely Kiss
  • Patricia Szokol
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Abstract

We prove continuity for reflexive, bisymmetric binary operations that are strictly increasing in each variable, under certain algebraic conditions. In fact, for some algebraic weights we construct a weighted quasi-arithmetic representation of such functions without assuming continuity. An elementary construction gives the case r=2/3. More generally, a word equality whose polynomial difference has a unique simple zero r∈(0,1) forces the representation with weight r; every algebraic r with no other conjugate in (0,1) arises this way. The proof uses an affine embedding and a dense-domain extension argument.

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Cite

BibTeXRIS
BibTeX
@misc{burai2026constructive,
  title = {{Constructive algebraic methods for bisymmetric means}},
  author = {Pál Burai and Gergely Kiss and Patricia Szokol},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.10021},
  eprint = {2611.10021},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.10021},
}
RIS
TY  - UNPB
TI  - Constructive algebraic methods for bisymmetric means
AU  - Pál Burai
AU  - Gergely Kiss
AU  - Patricia Szokol
PY  - 2026
DA  - 2026/11/04
PB  - arXiv
DO  - 10.48550/arxiv.2611.10021
UR  - https://arxiv.org/abs/2611.10021
AB  - We prove continuity for reflexive, bisymmetric binary operations that are strictly increasing in each variable, under certain algebraic conditions. In fact, for some algebraic weights we construct a weighted quasi-arithmetic representation of such functions without assuming continuity. An elementary construction gives the case r=2/3. More generally, a word equality whose polynomial difference has a unique simple zero r∈(0,1) forces the representation with weight r; every algebraic r with no other conjugate in (0,1) arises this way. The proof uses an affine embedding and a dense-domain extension argument.
ER  -