Autotopy
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 Albert (1943), based on his slightly earlier definition of isotopy for algebras, which was in turn inspired by work of Steenrod.
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.
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