In this paper, as a generalization of Urquhart's formulas, we present a full description of the sets of inner inverses and $(B,C)$-inverses over an arbitrary field. In addition, identifying the matrix-vector space with an affine space, we analyse geometrical properties of the main generalized inverse sets. We prove that the set of inner inverses, and the set of $(B,C)$-inverses, form affine subspaces and we study their dimensions. Furthermore, under some hypotheses, we prove that the set of outer inverses is not an affine subspace, but it is an affine algebraic variety. We also provide lower and upper bounds for the dimension of the outer inverse set.

中文翻译：

---

场上广义逆的表示和几何性质

在本文中，作为厄克特公式的推广，我们对内逆集和$(乙,C)$- 对任意场进行反演。此外，通过仿射空间识别矩阵向量空间，我们分析了主要广义逆集的几何性质。我们证明了内逆集和$(乙,C)$-逆，形成仿射子空间，我们研究它们的维度。此外，在一些假设下，我们证明了外逆集不是仿射子空间，而是仿射代数簇。我们还为外部逆集的维度提供下限和上限。