Back and Forth Systems of Condensations

TL;DR

This paper characterizes the relationships and equivalences between relational structures using back and forth systems, and explores their implications for reversibility and categoricity in model theory.

Contribution

It introduces new characterizations of condensability, bi-condensability, and reversibility of structures via back and forth systems and games, extending classical theorems.

Findings

Characterization of the preorder of condensability.

Hierarchy of structure similarities.

Homogeneous universal posets are not reversible.

Abstract

If $L$ is a relational language, an $L$-structure ${\mathbb X}$ is condensable to an $L$-structure ${\mathbb Y}$, we write ${\mathbb X} \preccurlyeq _c {\mathbb Y}$, iff there is a bijective homomorphism (condensation) from ${\mathbb X}$ onto ${\mathbb Y}$. We characterize the preorder $\preccurlyeq _c$, the corresponding equivalence relation of bi-condensability, ${\mathbb X} \sim _c {\mathbb Y}$, and the reversibility of $L$-structures in terms of back and forth systems and the corresponding games. In a similar way we characterize the ${\mathcal P}_{\infty \omega}$-equivalence (which is equivalent to the generic bi-condensability) and the ${\mathcal P}$-elementary equivalence of $L$-structures, obtaining analogues of Karp's theorem and the theorems of Ehrenfeucht and Fra\"iss\'e. In addition, we establish a hierarchy between the similarities of structures considered in the paper. Applying these results we show that homogeneous universal posets are not reversible and that a countable $L$-structure ${\mathbb X}$ is weakly reversible (that is, satisfies the Cantor-Schr\"oder-Bernstein property for condensations) iff its ${\mathcal P}_{\infty \omega}\cup {\mathcal N}_{\infty \omega}$-theory is countably categorical.