Semigroups in Distributed Computations : n-ary Operations and Irreducibility

This work extends the algebraic study of semigroups in distributed computation with focus on optimization, robustness, and higher-arity operations. We analyze pruning algorithms for discretized operator semigroups, yielding minimal generators that reduce redundancy and improve efficiency in distributed dataflows. Error analysis is developed through the concept of approximate semigroups, providing stability bounds for floating-point reductions under parallel aggregation. We examine canonical reduction rules, homomorphism-based optimizations, and algebraic compression techniques such as modular reduction. A key theme is the distinction between algebraic reducibility and practical efficiency: although $n$-ary laws can often embed into binary semigroups, distributed cost models highlight cases where native $n$-ary operators are irreducible and more suitable. Case studies including polynomial aggregation, median, majority, and determinants illustrate how categorical insights guide practical implementation strategies in systems like Spark and MapReduce.