岩浆操作数的商：格点结构和收敛重写系统
Quotients of the magmatic operad: lattice structures and convergent rewrite systems
摘要：
本文研究了岩浆算符，即一个二元生成元上的自由非对称算符的等价性。在线性设置中，我们表明，这些concurents的集合承认一个晶格结构，我们显示了一个类似的格拉斯曼公式的尺寸，这些operads。在非线性设置中，我们定义了梳状结合运算，这是由非负整数索引的运算推广的结合运算。我们证明了梳形结合运算集具有格结构，同构于具有除法阶的非负整数格。在计算机实验的驱动下，我们给出了对应于~ $3$的梳状结合运算的有限收敛表示.最后，通过一个三次关系研究了岩浆算符的导数，给出了它们的Hilbert级数表示和组合实现。
英文摘要：
We study quotients of the magmatic operad, that is the free nonsymmetric operad over one binary generator. In the linear setting, we show that the set of these quotients admits a lattice structure and we show an analog of the Grassmann formula for the dimensions of these operads. In the nonlinear setting, we define comb associative operads, that are operads indexed by nonnegative integers generalizing the associative operad. We show that the set of comb associative operads admits a lattice structure, isomorphic to the lattice of nonnegative integers equipped with the division order. Driven by computer experimentations, we provide a finite convergent presentation for the comb associative operad in correspondence with~$3$. Finally, we study quotients of the magmatic operad by one cubic relation by expressing their Hilbert series and providing combinatorial realizations.
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