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 nnLab.

Idea

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

Definition

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

ξ(a)η(b)=ζ(ab). \xi(a)\star \eta(b) = \zeta (a\cdot b).

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

Last revised on November 2, 2013 at 04:38:46. See the history of this page for a list of all contributions to it.