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

In a previous work the authors initiated a study on 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 extend the finite theory to parameterized and infinite families of mutations, drawing connections with quasispecies models in biology and interpreting image contractions as mechanisms of error suppression and genomic stability. By combining algebraic definitions, structural theorems, and algorithmic analyses, we provide a refined toolkit for understanding mutation collapse and its theoretical and biomedical implications.