Back and Forth Systems Witnessing Irreversibility

TL;DR

This paper characterizes the irreversibility of structures in relational languages using back and forth systems, and identifies classes of non-reversible structures including certain posets, lattices, ideals, and power sets.

Contribution

It introduces a new characterization of non-reversible structures via back and forth systems, enabling detection of non-reversibility in various classes of structures.

Findings

Identifies classes of non-reversible partial orders, including homogeneous-universal posets and the random poset.

Detects non-reversibility in divisibility lattices and ideals.

Provides set-theoretic conditions under which structures are non-reversible.

Abstract

If $L$ is a relational language, then an $L$-structure ${\mathbb X}=\langle X,\bar \rho \rangle$ is reversible iff there is no interpretation $\bar \sigma \varsubsetneq \bar \rho$ such that the structures $\langle X,\bar \sigma \rangle$ and $\langle X,\bar \rho \rangle$ are isomorphic. We show that ${\mathbb X}$ is not reversible iff there is a back and forth system $\Pi$ of partial self-condensations of ${\mathbb X}$ containing one which is not a partial isomorphism and having certain closure properties. Using that characterization we detect several classes of non-reversible partial orders containing, for example, homogeneous-universal posets (in particular, the random poset), the divisibility lattice, $\langle {\mathbb N} ,\,\mid\,\rangle$, the ideals $[\kappa ]^{<\lambda}$, the meager ideal in the algebra Borel$(\omega ^\omega)$, and the direct powers of rationals, ${\mathbb Q} ^\kappa$, and integers, ${\mathbb Z} ^\kappa$. Some of the results are obtained under additional set-theoretic assumptions.