## Hyers-Ulam stability of the Cauchy functional equation on square-symmetric groupoids.(English)Zbl 0980.39022

The author investigates the stability of the functional equation $f(x\diamond y)=f(x)\ast f(y) \text{ }(x,y \in X)$ 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 D. H. Hyers [Proc. Nat. Acad. Sci. USA 27, 222-224 (1941; Zbl 0061.26403)] on the stability of the Cauchy functional equation.

### MSC:

 39B72 Systems of functional equations and inequalities

### Keywords:

Hyers-Ulam stability

Zbl 0061.26403