Homotopy and semisymmetry of quasigroups
A quasigroup \({\mathcal Q}=(Q;\cdot,/,\setminus)\) is called semisymmetric if it satisfies the identity \((y\cdot x)y=x\); \(\mathcal Q\) is central if the diagonal \(\{(q,q);\;q\in Q\}\) is a normal subquasigroup of \({\mathcal Q}\times{\mathcal Q}\); \(\mathcal Q\) is entropic if it satisfies the identity \((xy)(zt)=(xz)(yt)\). By \(\mathbf{Qtp}\) is denoted the category of all quasigroups with homotopies between them and by \(\mathbf{Ztp}\) the category of central quasigroups with homotopies. Main results: The category \(\mathbf{Qtp}\) is isomorphic to a category of homomorphisms between semisymmetric quasigroups. The category \(\mathbf{Ztp}\) is isomorphic to a category of homomorphisms between entropic semisymmetric quasigroups.
scientific article; zbMATH DE number 1226234