A simple binary matroid is called claw-free if none of its rank-3 flats are independent sets. These objects can be equivalently defined as the sets E of points in $\mathrm{PG}(n-1,2)$ for which $E\cap P$ is not a basis of P for any plane P, or as the subsets X of ${\mathbb{F}}_2^n$ containing no linearly independent triple $x,y,z$ for which $x+y,y+z,x+z,x+y+z\notin X$.

We prove a decomposition theorem that exactly determines the structure of all claw-free matroids. The theorem states that claw-free matroids either belong to one of three particular basic classes of claw-free matroids, or can be constructed from these basic classes using a certain ‘join’ operation.

如果简单的二进制拟阵面中的第3级单位都不是独立的集合，则称为无爪类。这些对象可以被等效地定义为集合È在点$\mathrm{PG}（ñ-\mathrm{1个}，\mathrm{2个}）$ 为此 $E\cap P$不是的基P为任何平面P，或作为子集X的${\mathbb{F}}_{\mathrm{2个}}^{ñ}$ 不含线性独立的三元组 $X，ÿ，ž$ 为此 $X+ÿ，ÿ+ž，X+ž，X+ÿ+ž\notin X$。

我们证明了一个分解定理，该分解定理精确地确定了所有无爪类拟阵的结构。定理指出，无爪拟阵要么属于三种特定的无爪拟阵基本类之一，要么可以通过使用“联接”操作从这些基本类中构造出来。