作为树枝状代数推广的初始预代数
Initial pre-algebras as a generalization of dendriform algebras
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中文总结 AI 辅助
研究初始预代数，对二元运算律$\Var$定义初始预$\Var$ - 代数及运算律$\pre\Var^{\I}$，给出求定义关系的算法，研究自由初始预代数，还为非对称运算律$\pre\As^{\I}$构造基、描述线性基并建立双射。
中文摘要 AI 辅助
我们继续研究文献[DMS2026]中定义的初始双代数。对于二元运算律$\Var$，我们定义了初始预$\Var$ - 代数的类以及相应的运算律$\pre\Var^{\I}$，使得当$\Var$是二次时，有$(\di\Var^{\I})^{!}=(\pre(\Var^{!}))^{\I}$。我们提出一种直观算法来求当$\Var$是二元二次运算律时运算律$\pre\Var^{\I}$的定义关系。还研究了结合和交换情形下的自由初始预代数。对于非对称运算律$\pre\As^{\I}$，在自由岩浆运算律中构造了格罗贝纳 - 希尔绍夫基，用可允许装饰平面二叉树描述了线性基，并在这些树与某些组合对象之间建立了双射。
英文摘要
We continue the study of \emph{initial dialgebras} defined in~\cite{DMS2026}. For a binary operad $\Var$ we define the class of initial pre-$\Var$-algebras and the corresponding operad $\pre\Var^{\I}$ in such a way that \[ (\di\Var^{\I})^{!}=(\pre(\Var^{!}))^{\I} \] in the case when $\Var $ is quadratic. We propose an intuitive algorithm for finding the defining relations of the operad $\pre\Var^{\I}$ in the case when $\Var$ is a binary quadratic operad. We also study free initial pre-algebras in the associative and commutative settings. For the nonsymmetric operad $\pre\As^{\I}$, we construct a Grobner--Shirshov basis in the free magma operad, describe a linear basis in terms of admissible decorated planar binary trees, and establish a bijection between these trees and certain combinatorial objects.
