Hyers-Ulam stability of the Cauchy functional equation on square-symmetric groupoids (Q2754959)

The author investigates the stability of the functional equation NEWLINE\[NEWLINEf(x\diamond y)=f(x)\ast f(y) \text{ }(x,y \in X)NEWLINE\]NEWLINE where \(f:X \rightarrow Y\) and \((X, \diamond)\), \((Y, \ast)\) are groupoids with square-symmetric operations, i.e., operations \(\diamond\) and \(\ast\) satisfying \((x_1\diamond x_2)\diamond (x_1\diamond x_2)= (x_1\diamond x_1)\diamond (x_2 \diamond x_2)\) and \((y_1\ast y_2)\ast (y_1\ast y_2)= (y_1\ast y_1)\ast (y_2 \ast y_2)\) for all \(x_1,x_2 \in X\) and \(y_1,y_2 \in Y\), respectively. The results generalize the classical theorem of \textit{D. H. Hyers} [Proc. Nat. Acad. Sci. USA 27, 222-224 (1941; Zbl 0061.26403)] on the stability of the Cauchy functional equation.