Back and Forth Systems of Condensations

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.