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arXiv · 2608.19572

Disuccessors, Gr\"obner-Shirshov bases and free L-dendriform algebras

Huhu Zhang·Xing Gao·Yuanyuan Zhang
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Abstract

In this paper, we prove respectively that the disuccessor operations on the associative operad $\as$, the Lie operad $\lie$, and the pre-Lie operad $\prelie$ preserve Gr\"obner-Shirshov bases. This structural preservation enables the transfer of known bases to more complex operads. As a consequence, we introduce new methods for constructing Gr\"obner-Shirshov bases for the $\dend$ and $\prelie$ operads, and explicitly construct a Gr\"obner-Shirshov basis for the free L-dendriform algebra. This provides a conceptual and computationally efficient resolution of Madariaga's problem and offers new insights into the combinatorial structure of L-dendriform algebras.

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Huhu Zhang, Xing Gao, Yuanyuan Zhang. 2026-08-20. Disuccessors, Gr\"obner-Shirshov bases and free L-dendriform algebras. https://arxiv.org/abs/2608.19572

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