Isotopy of Loops - Isotopy of Quasigroups

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.