In 2004, Ehrenfeucht, Harju, and Rozenberg showed that any graph on a vertex set $V$ can be obtained from a complete graph on $V$ via a sequence of the operations of complementation, switching edges and non-edges at a vertex, and local complementation. The last operation involves taking the complement in the neighbourhood of a vertex. In this paper, we consider natural generalizations of these operations for binary matroids and explore their behaviour. We characterize all binary matroids obtainable from the binary projective geometry of rank $r$ under the operations of complementation and switching. Moreover, we show that not all binary matroids of rank at most $r$ can be obtained from a projective geometry of rank $r$ via a sequence of the three generalized operations. We introduce a fourth operation and show that, with this additional operation, we are able to obtain all binary matroids.

中文翻译：

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二元拟阵中的互补、局部互补和切换

2004 年，Ehrenfeucht、Harju 和 Rozenberg 证明了顶点集 $V$ 上的任何图都可以从 $V$ 上的完全图通过互补、在顶点处切换边和非边的一系列操作获得，和地方互补。最后一个操作涉及在顶点附近取补集。在本文中，我们考虑对二元拟阵进行这些操作的自然概括并探索它们的行为。我们描述了在互补和切换操作下可从秩 $r$ 的二元射影几何中获得的所有二元拟阵。此外，我们表明，并非所有秩最多为 $r$ 的二元拟阵都可以通过三个广义运算的序列从秩为 $r$ 的射影几何中获得。我们引入第四个操作并表明，通过这个额外的操作，