{
  "abstract": "The deterministic framework of mutation semigroups provides algebraic conditions for evolutionary collapse, but real mutation processes are inherently stochastic. This paper develops a comprehensive theory of stochastic mutation semigroups, where elementary mutations occur with empirically measured probabilities. A stochastic mutation semigroup is introduced as a Markov chain on the transformation semigroup generated by elementary mutation operators. We prove a stochastic transitivity threshold theorem under a positivity condition on the probability of rank reduction, correcting a logical gap in previous formulations, and we characterize collapse via the spectral properties of the associated Markov chain. Using HIV-1 sequence data from public databases and empirical mutation rates reported in the literature, the framework is illustrated through conceptual examples. Complete pseudocode is provided for the probabilistic pair-graph algorithm, along with convergence criteria and numerical examples demonstrating performance. The framework is further extended to infinite state spaces via topological semigroup theory and to time-varying mutation rates through dynamic L∗-classes. These results bridge the gap between abstract semigroup theory and evolutionary biology, offering a theoretical foundation for understanding stochastic mutation dynamics.",
  "abstract_shown": true,
  "affiliations": [
    "Department of Mathematics, Faculty of Physical Sciences, Akwa Ibom State University, Ikot Akpaden 534111, Nigeria",
    "Department of Mathematics, University of Calabar, Calabar 540281, Nigeria",
    "Department of Mathematics, College of Science and Humanities in Alkharj, Prince Sattam bin Abdulaziz University, Alkharj 11942, Saudi Arabia",
    "Department of Health and Rehabilitation Sciences, College of Applied Medical Sciences, Prince Sattam bin Abdulaziz University, Alkharj 11942, Saudi Arabia"
  ],
  "article_type": null,
  "arxiv": null,
  "author_total": 4,
  "authors": [
    {
      "affs": [
        0
      ],
      "name": "Marshal I. Sampson"
    },
    {
      "affs": [
        1
      ],
      "name": "Christiana F. Igiri"
    },
    {
      "affs": [
        2
      ],
      "name": "Reny George"
    },
    {
      "affs": [
        3
      ],
      "name": "Julie S. George"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e3tdjvgnbf203.bib",
  "cited_by": 0,
  "doi": "10.3390/mca31050195",
  "fulltext_url": "https://www.mdpi.com/2297-8747/31/5/195",
  "id": "e3tdjvgnbf203",
  "issns": [
    "2297-8747",
    "1300-686X"
  ],
  "landing_url": "https://doi.org/10.3390/mca31050195",
  "language": "en",
  "licence": "CC-BY",
  "licence_label": "CC BY",
  "licence_source": "journal",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "lcc",
      "QA1-939"
    ],
    [
      "lcc",
      "QA75.5-76.95"
    ],
    [
      "lcc",
      "T57-57.97"
    ],
    [
      "openalex",
      "1703|semigroups and automata theory"
    ]
  ],
  "openalex": "W7213952388",
  "pdf_url": null,
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": "MDPI AG",
  "ris": "https://research.draconitic.com/paper/e3tdjvgnbf203.ris",
  "sources": [
    {
      "first_seen": "2026-10-31T14:21:33Z",
      "native_id": "01b4dd24b765443cb8ba819c44341d88",
      "source": "doaj",
      "url": "https://doi.org/10.3390/mca31050195"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "lcc:QA1-939",
      "subfield": "general"
    }
  ],
  "title": "Stochastic Mutation Semigroups: From Deterministic Collapse to Probabilistic Evolutionary Dynamics",
  "title_text": "Stochastic Mutation Semigroups: From Deterministic Collapse to Probabilistic Evolutionary Dynamics",
  "type": "peer-reviewed",
  "updated_at": "2026-10-31T14:21:33Z",
  "url": "https://research.draconitic.com/paper/e3tdjvgnbf203",
  "venue": "Mathematical and Computational Applications",
  "versions": []
}