Computable Isomorphisms for Certain Classes of Infinite Graphs

TL;DR

This paper studies (2,1):1 structures, a class of infinite graphs with specific mapping properties, analyzing their computability features and providing conditions for when they are computably categorical.

Contribution

It introduces a sufficient condition for computable categoricity of (2,1):1 structures and explores their computability-theoretic properties with examples.

Findings

A sufficient condition for computable categoricity is established.

Examples demonstrate diverse computability properties of (2,1):1 structures.

The study extends understanding of infinite directed graphs in computability theory.

Abstract

We investigate (2,1):1 structures, which consist of a countable set $A$ together with a function $f: A \to A$ such that for every element $x$ in $A$, $f$ maps either exactly one element or exactly two elements of $A$ to $x$. These structures extend the notions of injection structures, 2:1 structures, and (2,0):1 structures studied by Cenzer, Harizanov, and Remmel, all of which can be thought of as infinite directed graphs. We look at various computability-theoretic properties of (2,1):1 structures, most notably that of computable categoricity. We say that a structure $\mathcal{A}$ is computably categorical if there exists a computable isomorphism between any two computable copies of $\mathcal{A}$. We give a sufficient condition under which a (2,1):1 structure is computably categorical, and present some examples of (2,1):1 structures with different computability-theoretic properties.