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：本文分析了乘法与取模运算的格线性性质。我们证明这两种运算具有格线性特性，并且针对这两种运算研究出的并行处理算法能够有效利用其相应问题的格线性结构。这表明此类算法可在异步环境中实现——允许节点间读取陈旧信息，但仍能保证在相同的时间复杂度内收敛。这些算法还表现出类似快照稳定性（snap-stabilization）的特性，即从任意初始状态出发，系统严格遵循规范规定的轨迹运行。