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Ultrahomogeneity and $ω$-categoricity of monounary algebras

Thomas Quinn-Gregson

Abstract

Ultrahomogeneity and $ω$-categoricity are two central concepts arising from model theory, with strong connections with oligomorphic permutation groups and quantifier elimination. In particular, both are conditions on the automorphism group of a structure. The aim of this paper is to describe both the $ω$-categorical monounary algebras and the ultrahomogeneous monounary algebras of arbitrary cardinalities. We show that a monounary algebra is $ω$-categorical [ultrahomogeneous] if and only if every element has finite height and Aut$(\mathcal{A})$ has only finitely many 1-orbits [$\mathcal{A}$ is 1-ultrahomogeneous]. Our classification of ultrahomogeneous monounary algebras is then viewed in the context of previously studied variants of ultrahomogeneity, including (partial)-homogeneity and transitivity.

Ultrahomogeneity and $ω$-categoricity of monounary algebras

Abstract

Ultrahomogeneity and -categoricity are two central concepts arising from model theory, with strong connections with oligomorphic permutation groups and quantifier elimination. In particular, both are conditions on the automorphism group of a structure. The aim of this paper is to describe both the -categorical monounary algebras and the ultrahomogeneous monounary algebras of arbitrary cardinalities. We show that a monounary algebra is -categorical [ultrahomogeneous] if and only if every element has finite height and Aut has only finitely many 1-orbits [ is 1-ultrahomogeneous]. Our classification of ultrahomogeneous monounary algebras is then viewed in the context of previously studied variants of ultrahomogeneity, including (partial)-homogeneity and transitivity.

Paper Structure

This paper contains 18 sections, 47 theorems, 20 equations.

Table of Contents

  1. Introduction
  2. Model theory preliminaries
  3. First order structures
  4. Permutation groups and orbit growth
  5. $\omega$-categoricity and definability
  6. Ultrahomogeneity
  7. Monounary algebras preliminaries
  8. The relational form
  9. Isomorphisms between monounary algebras
  10. Ultrahomogeneous monounary algebras
  11. Links to other variants of homogeneity
  12. Homogeneity
  13. Partial homogeneity
  14. The poset of variants of homogeneity
  15. Ultrahomogeneous directed pseudoforests
  16. ...and 3 more sections

Key Result

Corollary 1

For the class of monounary algebras we have

Theorems & Definitions (85)

  • Corollary
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4: Ryll-Nardzewski Theorem (RNT)
  • Definition 2.5
  • Corollary 2.6
  • Lemma 2.7
  • Definition 2.8
  • Theorem 2.9: Fraïssé's Theorem
  • ...and 75 more