Isotopy (in algebra)

For other meanings of the term see the disambiguation page Isotopy.

A relation on the class of all magmas defined by binary operations on a given set $M$. Namely, two operations $(M,{\cdot})$ and $(M,{\circ})$ are called isotopic if there exist permutations $\rho$, $\sigma$ and $\tau$ of $M$ such that for any $a,b\in M$, $$ a\circ b=(a\rho\cdot b\sigma)\tau \ . $$

The isotopy relation is an equivalence relation for the binary operations on $M$. An isomorphism of two binary operations defined on the same set is a special case of an isotopy (with $\rho=\sigma=\tau^{-1}$). An isotopy is called principal if $\tau$ is the identity permutation. Every isotope (i.e. isotopic magma) of a magma is isomorphic to a principal isotope of the magma. Every magma that is isotopic to a quasi-group is itself a quasi-group. Every quasi-group is isotopic to some loop (Albert's theorem). If a loop (in particular, a group) is isotopic to some group, then they are isomorphic. If a magma with identity is isotopic to a semi-group, then they are isomorphic, that is, they are both semi-groups with identity.