Computability and Categoricity of Weakly Ultrahomogeneous Structures

TL;DR

This paper explores the effective categoricity of ultrahomogeneous and weakly ultrahomogeneous structures, providing characterizations and complexity analyses for various classes such as linear orders and trees.

Contribution

It introduces the concept of weakly ultrahomogeneous structures, characterizes their properties, and compares them with known notions of computable and Δ^0_2 categoricity.

Findings

Computable ultrahomogeneous structures are Δ^0_2 categorical.

Characterizations for weakly ultrahomogeneous linear orders, equivalence structures, and trees are provided.

Complexity of ultrahomogeneity notions varies across different structure families.

Abstract

This paper investigates the effective categoricity of ultrahomogeneous structures. It is shown that any computable ultrahomogeneous structure is $\Delta^0_2$ categorical. A structure A is said to be weakly ultrahomogeneous if there is a finite (exceptional) set of elements $a_1,\dots,a_n$ such that A becomes ultrahomogeneous when constants representing these elements are added to the language. Characterizations are obtained for weakly ultrahomogeneous linear orderings, equivalence structures, injection structures and trees, and these are compared with characterizations of the computably categorical and $\Delta^0_2$ categorical structures. Index sets are used to determine the complexity of the notions of ultrahomegenous and weakly ultrahomogeneous for various families of structures.