On Shavrukov's non-isomorphism theorem for diagonalizable algebras

TL;DR

This paper strengthens Shavrukov's non-isomorphism theorem for diagonalizable algebras of certain theories, providing new examples and exploring bimodal cases, thereby advancing understanding of algebraic structures in logic.

Contribution

It presents a strengthened non-isomorphism theorem for diagonalizable algebras and explores bimodal cases with new examples of isomorphic and non-isomorphic pairs.

Findings

Proved a strengthened version of Shavrukov's non-isomorphism theorem.

Identified cases of epimorphisms between diagonalizable algebras.

Provided examples of isomorphic and non-isomorphic bimodal diagonalizable algebras.

Abstract

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