From Local Mutations to Global Fixation: A Semigroup Approach to Evolutionary Collapse

In a previous paper the authors initiated a study of mutation semigroups, where elementary mutation operations were encoded as total maps on finite sets and analyzed through structural, algebraic, and computational methods. Here we address several of the open problems raised therein. First, we investigate the algebraic characterization of generator sets that force the existence of constant or low-rank maps, linking these conditions to classical results on synchronizing automata. Second, we analyze the computational complexity of contraction-based heuristics, identifying cases where polynomial-time criteria are achievable and others where hardness results emerge. Finally, we discuss connections with quasispecies models in biology and interpret image contractions as mechanisms of error suppression and genomic stability, while noting that rigorous extension to infinite state spaces remains future work. By combining algebraic definitions, structural theorems, and algorithmic analyses, we provide a refined toolkit for understanding mutation collapse and its theoretical implications, with potential applications that require empirical validation beyond the scope of this paper.