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. Isotopy of quasigroups. Each quasigroup is isotopic to a loop. Let formula_1 and formula_2 be quasigroups. A quasigroup homotopy from "Q" to "P" is a triple 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 is a bijection. Two quasigroups are isotopic if there is an isotopy between them. In terms of Latin squares, an isotopy 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 formula_1 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. Isotopy of loops. Let formula_5 and formula_6 be loops and let formula_7 be an isotopy. Then it is the product of the principal isotopy formula_8 from formula_5 and formula_10 and the isomorphism formula_11 between formula_10 and formula_6. Indeed, put formula_14, formula_15 and define the operation formula_16 by formula_17. Let formula_5 and formula_19 be loops and let "e" be the neutral element of formula_5. Let formula_21 a principal isotopy from formula_5 to formula_19. Then formula_24 and formula_25 where formula_26 and  formula_27. A loop "L" is a G-loop if it is isomorphic to all its loop isotopes. Pseudo-automorphisms of loops. 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. Universal properties. 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. However, associativity and being an abelian group are universal properties. In fact, every group is a G-loop. The geometric interpretation of isotopy. Given a loop "L", one can define an incidence geometric structure called a 3-net. Conversely, after fixing an origin and an order of the line classes, a 3-net gives rise to a loop. Choosing a different origin or exchanging the line classes may result  in nonisomorphic coordinate loops. However, the coordinate loops are always isotopic. In other words, two loops are isotopic if and only if they are equivalent from "geometric point of view". The dictionary between algebraic and geometric concepts is as follows