Isotopy

Abstract

Let (G, •) and (H, o) be two groupoids. An ordered triple (α, β, γ) of one-to-one mappings α, β, γ of G upon H is called an isotopism of (G, •) upon (H, o), and (G, •) is said to be isotopic to or an isotope of (H, o), provided

$$ \left( {x\alpha } \right)o\left( {y\beta } \right) = \left( {x \cdot y} \right)\gamma $$

((1.1))

for all x, y in G. Isotopy of groupoids is clearly an equivalence relation. Moreover, given the groupoid (G, •) and one-to-one mappings α, β, γ of G upon a set H, (1.1) defines a groupoid (H, o) isotopic to (G, •). The element y is right nonsingular (Chapter II, § 9) in (G, •) if and only if y β is right nonsingular in (H, o); and similarly for left non-singularity. In particular, every isotope of a quasigroup is a quasigroup.