{"paper":{"title":"Irreducibility of Semigroup Morphisms","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"cs.FL","authors_text":"Daniel Reidenbach, Eva Foster, Paul C. Bell","submitted_at":"2026-03-16T12:11:52Z","abstract_excerpt":"We introduce and study the notion of irreducibility of semigroup morphisms over a finite alphabet. Given an alphabet $\\Sigma$, a morphism $\\varphi:\\Sigma^+\\rightarrow\\Sigma^+$ is irreducible if any factorisation $\\varphi=\\psi_2\\circ\\psi_1$ can only be satisfied if $\\psi_1$ or $\\psi_2$ is a trivial morphism; otherwise, $\\varphi$ is reducible. This definition provides a notion of primality in the endomorphism monoid of the free semigroup -- a natural and fundamental concept in this algebraic structure. The concept of the irreducibility of a morphism is related to the classical theory of simplifi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2603.15177","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2603.15177/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}