The aim of this paper is to extend the notions of geometric lattices, semimodularity and matroids in the framework of finite posets and related systems of sets. We define a geometric poset as one which is atomistic and which satisfies particular conditions connecting elements to atoms. Next, by using a suitable partial closure operator and the corresponding partial closure system, we define a partial matroid. We prove that the range of a partial matroid is a geometric poset under inclusion, and conversely, that every finite geometric poset is isomorphic to the range of a particular partial matroid. Finally, by introducing a new generalization of semimodularity from lattices to posets, we prove that a poset is geometric if and only if it is atomistic and semimodular.

本文的目的是在有限姿态和相关集系统的框架内扩展几何格，半模数和拟阵的概念。我们将几何姿态定义为原子态，并且满足将元素连接到原子的特定条件。接下来，通过使用合适的部分闭合运算符和相应的部分闭合系统，我们定义了部分拟阵。我们证明了部分拟阵的范围是包含条件下的几何姿态，相反，每个有限的几何阵对同一个特定拟阵的范围是同构的。最后，通过引入从晶格到波姿的半模态的新概括，我们证明了波姿是几何的，当且仅当它是原子且半模态的。