isotope of a groupoid
Let $G,H$ be groupoids^{} (http://planetmath.org/Groupoid). An isotopy^{} $\varphi $ from $G$ to $H$ is an ordered triple: $\varphi =(f,g,h)$, of bijections from $G$ to $H$, such that
$$f(a)g(b)=h(ab)\mathit{\hspace{1em}\hspace{1em}}\text{for all}a,b\in G.$$ 
$H$ is called an isotope of $G$ (or $H$ is isotopic to $G$) if there is an isotopy $\varphi :G\to H$.
Some easy examples of isotopies:

1.
If $f:G\to H$ is an isomorphism^{}, $(f,f,f):G\to H$ is an isotopy. By abuse of language^{}, we write $f=(f,f,f)$. In particular, $({1}_{G},{1}_{G},{1}_{G}):G\to G$ is an isotopy.

2.
If $\varphi =(f,g,h):G\to H$ is an isotopy, then so is
$${\varphi}^{1}:=({f}^{1},{g}^{1},{h}^{1}):H\to G,$$ 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.
If $\varphi =(f,g,h):G\to H$ and $\gamma =(r,s,t):H\to K$ are isotopies, then so is
$$\gamma \circ \varphi :=(r\circ f,s\circ g,t\circ h):G\to K,$$ for $(r\circ f)(a)(s\circ g)(b)=r(f(a))s(g(b))=t(f(a)g(b))=t(h(ab))=(t\circ h)(ab)$.
From the examples above, it is easy to see that “groupoids being isotopic” on the class of groupoids is an equivalence relation^{}, and that an isomorphism class is contained in an isotopic class. In fact, the containment is strict. For an example of nonisomorphic isotopic groupoids, see the reference below. However, if $G$ is a groupoid with unity and $G$ is isotopic to a semigroup^{} $S$, then it is isomorphic to $S$. Other conditions making isotopic groupoids isomorphic can be found in the reference below.
An isotopy of the form $(f,g,{1}_{H}):G\to H$ is called a principal isotopy, where ${1}_{H}$ 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 $G$.
Proof.
Suppose $(f,g,h):G\to H$ is an isotopy. To construct $K$, start with elements of $G$, which will form the underlying set of $K$. The binary operation^{} on $K$ is defined by
$$a\cdot b:=({f}^{1}\circ h)(a)({g}^{1}\circ h)(b).$$ 
Then $\cdot $ is welldefined, since $f,g$ are bijective^{}, for all pairs of elements of $G$. Hence $K$ is a groupoid. Furthermore, $({f}^{1}\circ h,{g}^{1}\circ h,{1}_{K}):G\to K$ is an isotopy by definition, so that $K$ is a principal isotope of $G$. Finally, $h(a\cdot b)=h({f}^{1}(h(a)){g}^{1}(h(b)))=f({f}^{1}(h(a)))g({g}^{1}(h(b)))=h(a)h(b)$, showing that $h:K\to H$ is a bijective homomorphism^{}, 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 proposition^{} suggests:
Proposition 1.
Any isotope of a quasigroup is a quasigroup.
Proof.
Suppose $(f,g,h):G\to H$ is an isotopy, and $G$ a quasigroup. Pick $x,z\in H$. Let $a,c\in G$ be such that $f(a)=x$ and $h(c)=z$. Let $b\in G$ be such that $ab=c$. Set $y=g(b)\in H$. Then $xy=f(a)g(b)=h(ab)=h(c)=z$. Similarly, there is $t\in H$ 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 $L$.
References
 1 R. H. Bruck: A Survey of Binary Systems. SpringerVerlag. New York (1966).
Title  isotope of a groupoid 
Canonical name  IsotopeOfAGroupoid 
Date of creation  20130322 18:35:54 
Last modified on  20130322 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  homotopy^{} 
Defines  homotope 
Defines  isotopic 
Defines  homotopic^{} 
Defines  principal isotopy 
Defines  principal isotope 