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We prove a strengthened version of V. Yu. Shavrukov's result on thenon-isomorphism of diagonalizable algebras of two $\Sigma_1$-sound theories,based on the improvements previously found by G. Adamsson. We then obtainseveral corollaries to the strengthened result by applying it to various pairsof theories and obtain new non-isomorphism examples. In particular, we showthat there is epimorphism from $(\mathfrak{L}_T, \Box_T \Box_T)$ onto$(\mathfrak{L}_T, \Box_T)$. The case of bimodal diagonalizable algebras is alsoconsidered. We give several examples of pairs of theories with isomorphicdiagonalizable algebras but non-isomorphic bimodal diagonalizable algebras.

On Shavrukov's non-isomorphism theorem for diagonalizable algebras