It is well-known that the lattice of all submodules of a module is modular. However, this is not the case for the lattice of subsemimodules of a semimodule. We show examples and describe these lattices for a given semimodule. We study closed and splitting subsemimodules and submodules of a given semimodule or module $\mathbf{M}$, respectively. We derive a sufficient condition under which the lattice ${\mathbf{L}}_c\left(\mathbf{M}\right)$ of closed subsemimodules is a homomorphic image of the lattice $\mathbf{L}\left(\mathbf{M}\right)$ of all subsemimodules. We describe the ordered set of splitting submodules of a module and show a natural bijective correspondence between this poset and the poset of all projections of this module. We show that this poset is orthomodular. This result extends the case known for the poset of closed subspaces of a Hilbert space which is used in the logic of quantum mechanics.

众所周知，一个模块的所有子模块的格都是模块化的。然而，对于半模的子半模的晶格，情况并非如此。我们展示示例并描述给定半模的这些格。我们研究给定半模块或模块的封闭和分裂子半模块和子模块$\mathbf{米}$， 分别。我们推导出一个充分条件，在该条件下，晶格${\mathbf{大号}}_C\left(\mathbf{米}\right)$的闭子半模是晶格的同态图像$\mathbf{大号}\left(\mathbf{米}\right)$所有子半模块。我们描述了一个模块的分裂子模块的有序集，并展示了这个poset和这个模块的所有投影的poset之间的自然双射对应。我们证明这个poset是正交模的。这个结果扩展了已知的用于量子力学逻辑中的希尔伯特空间的封闭子空间的偏集的情况。