Semigroups in Distributed Computations: Algebraic Foundations and Models

Abstract

This work develops algebraic foundations connecting semigroup theory with large-scale distributed computation. Classical constructions are revisited and extended by introducing metric and perturbed semigroups suited to modeling numerical processes. We present semigroup-based models for distributed aggregation, emphasizing Spark primitives and the limitations of binary reduction for inherently n-ary operations. Error propagation is treated through the framework of error semigroups, leading to robustness criteria that quantify resilience under perturbations. Case studies including Word Count, PageRank, and distributed matrix multiplication illustrate how algebraic structure governs both efficiency and reliability in computation.