Some varieties of groupoids which consist of Abelian group or group isotopes (Q1058076)

The identity \(W_ 1=W_ 2\) is balanced if each variable appears exactly twice in the identity, once on each side. The author proves by using the Reidemeister and other conditions that if \(W_ 1=W_ 2\) is a balanced identity of the form \(\_XY\_=\_TY\_\) such that there is at least one variable in the term T which does not appear in the term X, then every quasigroup (Q,\(\cdot)\) satisfying this identity is isotopic to an Abelian group. If similar identities are satisfied by certain cancellative groupoids then the Reidemeister or other closure conditions are fulfilled.