Semigroup Pruning Algorithms and Length Density in Numerical Semigroups

Semigroup Pruning Algorithms and Length Density in Numerical Semigroups | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Semigroup Pruning Algorithms and Length Density in Numerical Semigroups Marshal I. Sampson, Vivian Ndfutu Nfor, Christie Y. Ishola, Uwem P. Akai This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7694950/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract Length density δ(n) quantifies the proportion of attained factorization lengths of n within the interval [minL(n), maxL(n)] of a numerical semigroup. We derive explicit asymptotic lower bounds in terms of the minimal and maximal generators, proving that δ(n) → 1 uniformly as n → ∞. The analysis exploits extremal length estimates and introduces a general pruning template for factorization trees, from which both ordered and unordered pruning algorithms follow. We show that unordered pruning is valid only when atoms commute or when commutation relations ensure equivalence of permutations, while in non-commutative or non-cancellative settings pruning must be constrained by right-ideal and prefix-ACCP conditions. These results provide a unified framework connecting asymptotics, algorithmic pruning, and structural invariants such as elasticity and delta sets. Numerical experiments support the theoretical error bounds and confirm stabilization phenomena, demonstrating pruning as both an algebraic and computational tool in factorization theory. Pure Mathematics Numerical semigroups factorization theory length density pruning algorithms non-unique factorizations ACCP elasticity Full Text Additional Declarations The authors declare no competing interests. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-7694950","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":519438390,"identity":"bc42a090-50e9-4b00-81d6-5c721be4c54d","order_by":0,"name":"Marshal I. 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We derive explicit asymptotic lower bounds in terms of the minimal and maximal generators, proving that δ(n) → 1 uniformly as n → ∞. The analysis exploits extremal length estimates and introduces a general pruning template for factorization trees, from which both ordered and unordered pruning algorithms follow. We show that unordered pruning is valid only when atoms commute or when commutation relations ensure equivalence of permutations, while in non-commutative or non-cancellative settings pruning must be constrained by right-ideal and prefix-ACCP conditions. These results provide a unified framework connecting asymptotics, algorithmic pruning, and structural invariants such as elasticity and delta sets. 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