In Search of Homology for Quasigroups of Bol-Moufang Type

2025/08/29 by Anthony Christiana, Ben Clingenpeel, Christiana, Anthony +9
Computer Science · Mathematics · #20N05 #Advanced Algebra and Logic #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2508.21268

openalex publication_date 2025/08/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We initiate (co)homology theory for quasigroups of Bol-Moufang type based on analysis of their extensions by affine quasigroups of the same type. We use these extensions to define second and third boundary operations, ∂₂(x,y) and ∂₃(x,y,z), respectively. We use these definitions to compute the second homology groups for several examples from the work of Phillips and Vojtechovsky. We speculate about the relation between these homology groups and those obtained from a small category with coefficients in a functor.

Citations

• The second Yang-Baxter homology for the Homflypt polynomial
• Abelian quandles and quandles with abelian structure group
• Equivalence of two definitions of set-theoretic Yang-Baxter homology
• Knots and distributive homology: from arc colorings to Yang-Baxter homology
• Homologies of Algebraic Structures via Braidings and Quantum Shuffles
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Related

• Ternary quasigroups in knot theory
• Units in quasigroups with classical Bol-Moufang type identities
• Distributive and trimedial quasigroups of order 243
• Low-dimensional homology groups of mapping class groups: a survey
• Topological spaces related with topological quasigroups