Spherical orders, properties and countable spectra of their theories

TL;DR

This paper investigates the properties and spectra of theories related to spherical orders, demonstrating their categoricity, decidability, and confirming the Vaught conjecture for certain expansions.

Contribution

It provides a detailed analysis of the theories of spherical orders, including their spectra and model properties, with new results on categoricity and the Vaught conjecture.

Findings

Theories of dense n-spherical orders are countably categorical.

These theories are decidable.

The Vaught conjecture holds for countable constant expansions of dense n-spherical theories.

Abstract

We study semantic and syntactic properties of spherical orders and their elementary theories, including finite and dense orders and their theories. It is shown that theories of dense $n$-spherical orders are countably categorical and decidable. The values for spectra of countable models of unary expansions of $n$-spherical theories are described. The Vaught conjecture is confirmed for countable constant expansions of dense $n$-spherical theories.