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.