isotope of a groupoid

Let G,H be groupoidsPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath ( An isotopyMathworldPlanetmath ϕ from G to H is an ordered triple: ϕ=(f,g,h), of bijections from G to H, such that

f(a)g(b)=h(ab)  for all a,bG.

H is called an isotope of G (or H is isotopic to G) if there is an isotopy ϕ:GH.

Some easy examples of isotopies:

  1. 1.

    If f:GH is an isomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath, (f,f,f):GH is an isotopy. By abuse of languagePlanetmathPlanetmath, we write f=(f,f,f). In particular, (1G,1G,1G):GG is an isotopy.

  2. 2.

    If ϕ=(f,g,h):GH is an isotopy, then so is


    for if f-1(a)=c and g-1(b)=d, then ab=f(c)g(d)=h(cd), so that f-1(a)g-1(b)=cd=h-1(ab)

  3. 3.

    If ϕ=(f,g,h):GH and γ=(r,s,t):HK are isotopies, then so is


    for (rf)(a)(sg)(b)=r(f(a))s(g(b))=t(f(a)g(b))=t(h(ab))=(th)(ab).

From the examples above, it is easy to see that “groupoids being isotopic” on the class of groupoids is an equivalence relationMathworldPlanetmath, and that an isomorphism class is contained in an isotopic class. In fact, the containment is strict. For an example of non-isomorphic isotopic groupoids, see the reference below. However, if G is a groupoid with unity and G is isotopic to a semigroupPlanetmathPlanetmath S, then it is isomorphic to S. Other conditions making isotopic groupoids isomorphic can be found in the reference below.

An isotopy of the form (f,g,1H):GH is called a principal isotopy, where 1H is the identity function on H. H is called a principal isotope of G. If H is isotopic to G, then H is isomorphic to a principal isotope K of G.


Suppose (f,g,h):GH is an isotopy. To construct K, start with elements of G, which will form the underlying set of K. The binary operationMathworldPlanetmath on K is defined by


Then is well-defined, since f,g are bijectiveMathworldPlanetmath, for all pairs of elements of G. Hence K is a groupoid. Furthermore, (f-1h,g-1h,1K):GK is an isotopy by definition, so that K is a principal isotope of G. Finally, h(ab)=h(f-1(h(a))g-1(h(b)))=f(f-1(h(a)))g(g-1(h(b)))=h(a)h(b), showing that h:KH is a bijective homomorphismMathworldPlanetmathPlanetmathPlanetmath, and hence an isomorphism. ∎

Remark. In the literature, the definition of an isotope is sometimes limited to quasigroups. However, this is not necessary, as the follow propositionPlanetmathPlanetmath suggests:

Proposition 1.

Any isotope of a quasigroup is a quasigroup.


Suppose (f,g,h):GH is an isotopy, and G a quasigroup. Pick x,zH. Let a,cG be such that f(a)=x and h(c)=z. Let bG be such that ab=c. Set y=g(b)H. Then xy=f(a)g(b)=h(ab)=h(c)=z. Similarly, there is tH such that tx=z. Hence H is a quasigroup. ∎

On the other hand, an isotope of a loop may not be a loop. Nevertheless, we sometimes say that an isotope of a loop L as a loop isotopic to L.


  • 1 R. H. Bruck: A Survey of Binary Systems.  Springer-Verlag. New York (1966).
Title isotope of a groupoid
Canonical name IsotopeOfAGroupoid
Date of creation 2013-03-22 18:35:54
Last modified on 2013-03-22 18:35:54
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 8
Author CWoo (3771)
Entry type Definition
Classification msc 20N02
Classification msc 20N05
Synonym isotopism
Synonym homotopism
Defines isotopy
Defines isotope
Defines homotopyMathworldPlanetmath
Defines homotope
Defines isotopic
Defines homotopicMathworldPlanetmath
Defines principal isotopy
Defines principal isotope