Isotopy of loops

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. 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. Let and be loops and let be an isotopy. Then it is the product of the principal isotopy from and and the isomorphism between and . Indeed, put , and define the operation by . Let and be loops and let e be the neutral element of . Let a principal isotopy from to . Then and where and . A loop L is a G-loop if it is isomorphic to all its loop isotopes. Let L be a loop and c an element of L. A bijection α of L is called a right pseudo-automorphism of L with companion element c if for all x, y the identity holds. One defines left pseudo-automorphisms analogously. We say that a loop property P is universal if it is isotopy invariant, that is, P holds for a loop L if and only if P holds for all loop isotopes of L. Clearly, it is enough to check if P holds for all principal isotopes of L. For example, since the isotopes of a commutative loop need not be commutative, commutativity is not universal.