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) ⊗sub> T I(Y,S) → I(X x Y, R ⊗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] ⊗T S[H] ≅ (R ⊗T S)[G × H] is valid.

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