{"paper":{"title":"Convex spaces, affine spaces, and commutants for algebraic theories","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RA"],"primary_cat":"math.CT","authors_text":"Rory B. B. Lucyshyn-Wright","submitted_at":"2016-03-10T17:54:54Z","abstract_excerpt":"Certain axiomatic notions of $\\textit{affine space}$ over a ring and $\\textit{convex space}$ over a preordered ring are examples of the notion of $\\mathcal{T}$-algebra for an algebraic theory $\\mathcal{T}$ in the sense of Lawvere. Herein we study the notion of $\\textit{commutant}$ for Lawvere theories that was defined by Wraith and generalizes the notion of $\\textit{centralizer clone}$. We focus on the Lawvere theory of $\\textit{left $R$-affine spaces}$ for a ring or rig $R$, proving that this theory can be described as a commutant of the theory of pointed right $R$-modules. Further, we show t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.03351","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":""},"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"}