Huhu Zhang, Xing Gao, Yuanyuan Zhang
Abstract
In this paper, we prove respectively that the disuccessor operations on the associative operad , the Lie operad , and the pre-Lie operad preserve Gröbner-Shirshov bases. This structural preservation enables the transfer of known bases to more complex operads. As a consequence, we introduce new methods for constructing Gröbner-Shirshov bases for the and operads, and explicitly construct a Gröbner-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.