nLab isotope (algebra)

This entry is about the isotope and isotopy in algebra, for isotopy in geometry see isotopy; homotopy of quasigroups is also covered, do not mix with the notion from homotopy theory. For isotope in chemistry and nuclear physics see wikipedia; along with isotopic spin? might be once covered in $n$Lab.

Idea

Homotopy and isotopy are relaxed notions of homomorphism and isomorphism suitable for nonassociative binary algebraic structures.

Definition

Let $(A,\cdot), (B,\star)$ be binary algebraic structures (magmas) and $\xi,\eta,\zeta : A\to B$ set maps. The triple $(\xi,\eta,\zeta)$ is a homotopy of binary structures if for all $x,y\in A$

$$\xi(a)\star \eta(b) = \zeta (a\cdot b).$$

A homotopy is an isotopy if $\xi,\eta,\zeta$ are bijections of sets; if there is an isotopy $(A,\cdot)\to (B,\star)$, then we say that $(A,\cdot)$ is isotopic to $(B,\star)$, or that they are isotopes. Autotopy is an isotopy from $(A,\cdot)$ to itself.

Properties

Being isotopic is a relation of equivalence. Magmas and their isotopies from a groupoid.

Every loop isotopic to a group is isomorphic to a group. This is why isotopy is a non-interesting notion for groups. There exist a quasigroup which is isotopic to a group but not isomorphic to a group.

Literature

• R. Artzy, Isotopy an parastrophy of quasigroups, Proc. Amer. Math. Soc. 14 (1963)

category: algebra