Tensor product of incidence algebras and group algebras

Let $I(X, R)$ and $I(Y, S)$ be incidence algebras, where $X$ and $Y$ are preordered sets, $R$ and $S$ are algebras over some commutative ring $T$. We prove the existence of a homomorphism of algebras $I(X, R)\otimes_T I(Y, S)\to I(X\times Y, R\otimes_T S)$. If $X$ and $Y$ are finite sets, then there is an isomorphism. For arbitrary groups $G$ and $H$, it is proved that the isomorphism of algebras $R[G]\otimes_T S[H]\cong (R\otimes_T S)[G\times H]$ is valid.