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In the mathematical field of abstract algebra, isotopy is an equivalence relation used to classify the algebraic notion of loop. Isotopy for loops and quasigroups was introduced by , based on his slightly earlier definition of isotopy for algebras, which was in turn inspired by work of Steenrod. == Isotopy of quasigroups == Each quasigroup is isotopic to a loop. Let and be quasigroups. A quasigroup homotopy from ''Q'' to ''P'' is a triple (α, β, γ) of maps from ''Q'' to ''P'' such that : for all ''x'', ''y'' in ''Q''. A quasigroup homomorphism is just a homotopy for which the three maps are equal. An isotopy is a homotopy for which each of the three maps (α, β, γ) is a bijection. Two quasigroups are isotopic if there is an isotopy between them. In terms of Latin squares, an isotopy (α, β, γ) is given by a permutation of rows α, a permutation of columns β, and a permutation on the underlying element set γ. An autotopy is an isotopy from a quasigroup to itself. The set of all autotopies of a quasigroup form a group with the automorphism group as a subgroup. A principal isotopy is an isotopy for which γ is the identity map on ''Q''. In this case the underlying sets of the quasigroups must be the same but the multiplications may differ. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「isotopy of loops」の詳細全文を読む スポンサード リンク
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