부산대학교 도서관

Let $N \in \Bbb{N}$. When $X_{1}, X_{2}, \dots, X_{N}$ are sets, $\Cal{R}$ is a semigroup, and $f_{n}\:X_{n} \rightarrow \Cal{R}$ is a mapping for $n = 1, 2, \dots, N$, define $f_{1} \otimes f_{2} \otimes \cdots \otimes f_{N}\: X_{1} \times X_{2} \times \cdots \times X_{N} \rightarrow \Cal{R}$ by $$ (f_{1} \otimes f_{2} \otimes \cdots \otimes f_{N})(x_{1}, x_{2}, \dots, x_{N}) \coloneq f_{1}(x_{1}) f_{2}(x_{2}) \cdots f_{N}(x_{N}) $$ for $(x_{1}, x_{2}, \dots, x_{N}) \in X_{1} \times X_{2} \times \cdots \times X_{N}$. Now let $S_{1}, S_{2}, \dots, S_{N}$ and $\Cal{R}$ be semigroups. In this paper, for homomorphisms $F \: S_{1} \times S_{2} \times \cdots \times S_{N} \rightarrow \Cal{R}$, the author derives sufficient conditions for the existence and uniqueness of homomorphisms $F_{n}\: S_{n} \rightarrow \Cal{R}$, $n = 1, 2, \dots, N$, such that $$ F(s_{1}, s_{2}, \dots, s_{N}) = F_{1}(s_{1})F_{2}(s_{2}) \cdots F_{N}(s_{N}) $$ for all $s_{1} \in S_{1}, s_{2} \in S_{2}, \dots, s_{N} \in S_{N}$. The author then applies the theory to some interesting examples, looking at additive and multiplicative functions on a product of subsemigroups of $\Bbb{N}$, and fitting some functional equations in $\Bbb{R}^{2}$ into this framework in order to solve them.