Consider a Borcherds-Kac-Moody Lie superalgebra, denoted as $\mathfrak{g}$, associated with the graph G. This Lie superalgebra is constructed from a free Lie superalgebra by introducing three sets of relations on its generators: (1) Chevalley relations, (2) Serre relations, and (3) The commutation relations derived from the graph G.

The Chevalley relations lead to a triangular decomposition of $\mathfrak{g}$ as $\mathfrak{g}={\mathfrak{n}}_{+}\oplus \mathfrak{h}\oplus {\mathfrak{n}}_{-}$, where each root space ${\mathfrak{g}}_{\alpha }$ is contained in either ${\mathfrak{n}}_{+}$ or ${\mathfrak{n}}_{-}$. Importantly, each ${\mathfrak{g}}_{\alpha }$ is determined solely by relations (2) and (3). This paper focuses on the root spaces of $\mathfrak{g}$ that are unaffected by the Serre relations. We refer to these root spaces as “free roots” of $\mathfrak{g}$ (these root spaces are free from the Serre relations and can be associated with certain grade spaces of freely partially commutative Lie superalgebras, as detailed in Lemma 3.10. Consequently, we refer to them as “free roots,” and the corresponding root spaces in $\mathfrak{g}$ as “free root spaces” [cf. Definition 2.6]). Since these root spaces only involve commutation relations derived from the graph G, we can examine them purely from a combinatorial perspective.

We employ heaps of pieces to analyze these root spaces and establish various combinatorial properties. We develop two distinct bases for these root spaces of $\mathfrak{g}$: We extend Lalonde's Lyndon heap basis, originally designed for free partially commutative Lie algebras, to accommodate free partially commutative Lie superalgebras. We expand upon the basis introduced in the reference [1], designed for the free root spaces of Borcherds algebras, to encompass BKM superalgebras. This extension is achieved by investigating the combinatorial properties of super Lyndon heaps. Additionally, we also explore several other combinatorial properties related to free roots.

考虑 Borcherds-Kac-Moody Lie 超代数，表示为$\mathfrak{G}$，与图G相关联。该李超代数是通过在其生成器上引入三组关系从自由李超代数构造的：(1) Chevalley 关系，(2) Serre 关系，以及 (3) 从图 G 导出的交换关系。

Chevalley 关系导致三角分解$\mathfrak{G}$作为$\mathfrak{G}={\mathfrak{n}}_{+}\oplus \mathfrak{H}\oplus {\mathfrak{n}}_{-}$，其中每个根空间${\mathfrak{G}}_{\alpha }$包含在任一${\mathfrak{n}}_{+}$或者${\mathfrak{n}}_{-}$。重要的是，每个${\mathfrak{G}}_{\alpha }$仅由关系式（2）和（3）决定。本文主要研究根空间$\mathfrak{G}$不受 Serre 关系的影响。我们将这些根空间称为“自由根”$\mathfrak{G}$（这些根空间不受 Serre 关系的影响，并且可以与自由部分交换李超代数的某些等级空间相关联，如引理 3.10中详述。因此，我们将它们称为“自由根”，并且在$\mathfrak{G}$作为“自由根空间”[cf. 定义2.6 ]）。由于这些根空间仅涉及从图G导出的交换关系，因此我们可以纯粹从组合角度来检查它们。

我们使用大量的碎片来分析这些根空间并建立各种组合属性。我们为这些根空间开发了两个不同的基础$\mathfrak{G}$：我们扩展了 Lalonde 的 Lyndon 堆基础，最初是为自由部分交换李代数设计的，以适应自由部分交换李超代数。我们在参考文献[1]中介绍的基础上进行了扩展，该基础是为 Borcherds 代数的自由根空间设计的，以包含 BKM 超代数。这一扩展是通过研究超级林登堆的组合特性来实现的。此外，我们还探索了与自由根相关的其他几个组合特性。