Ternary analogs of dendriform algebras and quadri-algebras

Abstract

We construct ternary analogs of dendriform algebras and quadri-algebras, generalizing the well-known binary case. These new structures are defined in terms of nonsymmetric operads. We construct these new operads using ternary operations defined in terms of ternary Rota-Baxter operators. The relations defining these operads lie in the quadratic subspace of the kernel of a rewriting morphism from a free operad to the operad generated by one associative ternary operation and one (or two commuting) unary operators satisfying the ternary Rota-Baxter relation(s). We use computational algebra to compute a basis for the quadratic component of the kernel and thereby obtain the quadratic relations which define these new operads. In the dendriform case, we obtain 3 operations satisfying 10 relations. In the quadri-algebra case, we obtain 9 operations satisfying 50 relations. The latter computations combine modular and rational arithmetic, but we are able to provide certificates of the results which can be verified in characteristic 0.

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BibTeX RIS

BibTeX

@misc{hin2026ternary,
  title = {{Ternary analogs of dendriform algebras and quadri-algebras}},
  author = {Yu Hin and Au and Murray R. Bremner},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.37848},
  eprint = {2610.37848},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.37848},
}

RIS

TY  - UNPB
TI  - Ternary analogs of dendriform algebras and quadri-algebras
AU  - Yu Hin
AU  - Au
AU  - Murray R. Bremner
PY  - 2026
DA  - 2026/10/27
PB  - arXiv
DO  - 10.48550/arxiv.2610.37848
UR  - https://arxiv.org/abs/2610.37848
AB  - We construct ternary analogs of dendriform algebras and quadri-algebras, generalizing the well-known binary case. These new structures are defined in terms of nonsymmetric operads. We construct these new operads using ternary operations defined in terms of ternary Rota-Baxter operators. The relations defining these operads lie in the quadratic subspace of the kernel of a rewriting morphism from a free operad to the operad generated by one associative ternary operation and one (or two commuting) unary operators satisfying the ternary Rota-Baxter relation(s). We use computational algebra to compute a basis for the quadratic component of the kernel and thereby obtain the quadratic relations which define these new operads. In the dendriform case, we obtain 3 operations satisfying 10 relations. In the quadri-algebra case, we obtain 9 operations satisfying 50 relations. The latter computations combine modular and rational arithmetic, but we are able to provide certificates of the results which can be verified in characteristic 0.
ER  -

Abstract

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