Fuzzy \(G\)-equivalences and \(G\)-congruences on a groupoid under semibalanced maps (Q1808923)

In the present note we obtain some basic results for \(G\)-equivalences and \(G\)-congruences. For two nonempty sets \(X\) and \(Y\), we call a mapping \(f\colon X\times X\to Y\times Y\) a semibalanced map if (i) given \(a\in X\), there exists \(u\in Y\) such that \(f(a,a)=(u,u)\), and (ii) \(f(a,a)=(u,u)\), \(f(b,b)=(v,v)\) imply that \(f(a,b)=(u,v)\). We highlight the fact that the role of semibalanced maps for \(G\)-equivalences and \(G'\)-congruences is analogous to that of isomorphisms for classical algebraic structures.