In this paper, we analyze lattice linearity of multiplication and modulo operations. We demonstrate that these operations are lattice linear and the parallel processing algorithms that we study for both these operations are able to exploit the lattice linearity of their respective problems. This implies that these algorithms can be implemented in asynchronous environments, where the nodes are allowed to read old information from each other and are still guaranteed to converge within the same time complexity. These algorithms also exhibit properties similar to snap-stabilization, i.e., starting from an arbitrary state, the system follows the trace strictly according to its specification.

翻译：在本文中，我们分析了乘法和取模操作的格线性。我们证明了这些操作是格线性的，并且我们研究的并行处理算法能够利用它们各自的问题的格线性。这意味着这些算法可以在异步环境中实现，其中节点被允许从彼此读取旧信息，并且仍然保证在相同的时间复杂度内收敛。这些算法也表现出类似于快照稳定的性质，即从任意状态开始，系统严格按照其规范跟踪跟随。