above four identities has the inverse property and therefore satisfies all four. Any loop which satisfies the left, right, or antiautomorphic inverse properties automatically has two-sided inverses. Morphisms A quasigroup or loop homomorphism is a map between two quasigroups such that . Quasigroup homomorphisms necessarily preserve left and right division, as well as identity elements (if they exist). Homotopy and isotopy Let Q and P 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 to itself. The set of all autotopies of a quasigroup form a group with the automorphism group as a subgroup. Every quasigroup is isotopic to a loop. If a loop is isotopic to a group, then it is isomorphic to that group and thus is itself a group. However, a quasigroup which is isotopic to a group need not be a group. For example, the quasigroup on R with multiplication given by is isotopic to the additive group , but is not itself a group. Every medial quasigroup is isotopic to an abelian group by the Bruck–Toyoda theorem. Conjugation (parastrophe) Left and right division are examples of forming a quasigroup by permuting the variables in the defining equation. From the original operation ∗ (i.e., ) we can form five new operations: (the opposite operation), / and \, and their opposites. That makes a total of six quasigroup operations, which are called the conjugates or parastrophes of ∗. Any two of these operations are said to be "conjugate" or "parastrophic" to each other (and to themselves). Isostrophe (paratopy) If the set Q has two quasigroup operations, ∗ and ·, and one of them is isotopic to a conjugate of the other, the operations are said to be isostrophic to each other. There are also many other names for this relation of "isostrophe", e.g., paratopy. Generalizations Polyadic or multiary quasigroups An n-ary quasigroup is