Isomorphisms of incidence algebras

Let Y and X be preordered sets, R be an algebra over some commutative ring, K′ = I(Y, R) and K = I(X, R) be incidence algebras. Several questions can be formulated regarding isomorphisms between the algebras K′ and K. One of them is known as the isomorphism problem. It is usually written in the following form. If the algebras K′ and K are isomorphic, then will Y and X be isomorphic as preordered sets? Another general question asks us to find the structure of isomorphisms between K′ and K. The article contains two theorems. Theorem 3.1, under certain assumptions about the algebras K′ and K and the ring R, gives a positive answer to the isomorphism problem. Theorem 3.2, under one condition on the algebras K′ and K, states that any isomorphism of the algebras K′ and K after conjugation by an inner automorphism of the algebra K becomes a diagonal (in a certain sense) isomorphism.

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