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Expansions and restrictions of structures and theories, their hierarchies

Sergey V. Sudoplatov

TL;DR

This work develops a general framework to analyze how expansions and restrictions of structures and their theories interact with core model-theoretic properties, using regular structures and the Boolean algebra $\mathcal{B}(\mathcal{M})$ of all universe-preserving restrictions of a maximal regular expansion. It introduces notions of $E$- and $R$-properties, lattice structures, and permutation invariance, and applies the framework to $\ω$-categorical, Ehrenfeucht, strongly minimal, $ω_1$-categorical, and stable classes. It shows that properties like $ω$-categoricity, Ehrenfeuchtness, $ω_1$-categoricity, and stability can fail under fusion, while strongly minimal structures form a distributive sublattice $\mathcal{B}_{sm}(\mathcal{M})$ with a greatest element $SM$ and all restrictions of $SM$ remain strongly minimal. The results provide precise criteria for preservation of properties under expansions and restrictions and offer a systematic tool for studying hierarchical behavior across algebraic, geometric, and ordered theories.

Abstract

We introduce and study some general principles and hierarchical properties of expansions and restrictions of structures and their theories The general approach is applied to describe these properties for classes of $ω$-categorical theories and structures, Ehrenfeucht theories and their models, strongly minimal, $ω_1$-theories, and stable ones.

Expansions and restrictions of structures and theories, their hierarchies

TL;DR

This work develops a general framework to analyze how expansions and restrictions of structures and their theories interact with core model-theoretic properties, using regular structures and the Boolean algebra of all universe-preserving restrictions of a maximal regular expansion. It introduces notions of - and -properties, lattice structures, and permutation invariance, and applies the framework to -categorical, Ehrenfeucht, strongly minimal, -categorical, and stable classes. It shows that properties like -categoricity, Ehrenfeuchtness, -categoricity, and stability can fail under fusion, while strongly minimal structures form a distributive sublattice with a greatest element and all restrictions of remain strongly minimal. The results provide precise criteria for preservation of properties under expansions and restrictions and offer a systematic tool for studying hierarchical behavior across algebraic, geometric, and ordered theories.

Abstract

We introduce and study some general principles and hierarchical properties of expansions and restrictions of structures and their theories The general approach is applied to describe these properties for classes of -categorical theories and structures, Ehrenfeucht theories and their models, strongly minimal, -theories, and stable ones.

Paper Structure

This paper contains 10 sections, 17 theorems, 7 equations.

Table of Contents

  1. Introduction
  2. Regular structures, their expansions, restrictions and Boolean algebras
  3. Kinds of properties with respect to Boolean algebras
  4. Properties forming lattices
  5. Structures and properties closed under permutations
  6. The families of $\omega$-categorical and Ehrenfeucht structures in $\mathcal{B}(\mathcal{M})$
  7. The family of strongly minimal structures in $\mathcal{B}(\mathcal{M})$
  8. The family of $\omega_1$-categorical structures in $\mathcal{B}(\mathcal{M})$
  9. The family of stable structures in $\mathcal{B}(\mathcal{M})$
  10. Conclusion

Key Result

Theorem 2.2

For any regular structures $\mathcal{M}$ and $\mathcal{N}$ the following conditions are equivalent: $(1)$$\mathcal{B}(\mathcal{M})\simeq\mathcal{B}(\mathcal{N})$, $(2)$ there is a bijection between sets of atoms for $\mathcal{B}(\mathcal{M})$ and $\mathcal{B}(\mathcal{N})$, $(3)$${\rm max}\{|M|,\ome

Theorems & Definitions (36)

  • Remark 2.1
  • Theorem 2.2
  • Corollary 2.3
  • Theorem 2.4
  • Corollary 2.5
  • Remark 2.6
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Proposition 3.4
  • ...and 26 more