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：本文分析了乘法与模运算的格线性性质。我们证明这些运算具有格线性，并且针对这两种运算所研究的并行处理算法能够利用各自问题的格线性特性。这意味着这些算法可在异步环境中实现——允许节点读取彼此的旧信息，但仍保证在相同时间复杂度内收敛。此外，这些算法还表现出类似于快照稳定性的特性，即从任意初始状态开始，系统严格依据其规范执行轨迹。