Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Hierarchies of direct powers, ultrapowers and cumulative powers

Pedro Teixeira Yago

Abstract

In this paper we investigate cumulative hierarchies of functions on structures, or cumulative powers, and study their properties. Particularly, we show how they extend the preservation phenomena of reduced powers, direct powers and ultrapowers by offering a characterization result of the fragment of first-order theory it preserves, and elucidate the connection between the three sorts of constructions. More precisely, we show how both direct powers and ultrapowers may be obtained from cumulative powers as a quotient by appropriate equivalence relations. We address how embeddability lifts from generating structures to their cumulative powers, direct powers and ultrapowers, and under what conditions ultrapowers embed into corresponding cumulative powers or direct powers. We further offer an application of the framework to show a straightforward way of constructing a surreal field.

Hierarchies of direct powers, ultrapowers and cumulative powers

Abstract

In this paper we investigate cumulative hierarchies of functions on structures, or cumulative powers, and study their properties. Particularly, we show how they extend the preservation phenomena of reduced powers, direct powers and ultrapowers by offering a characterization result of the fragment of first-order theory it preserves, and elucidate the connection between the three sorts of constructions. More precisely, we show how both direct powers and ultrapowers may be obtained from cumulative powers as a quotient by appropriate equivalence relations. We address how embeddability lifts from generating structures to their cumulative powers, direct powers and ultrapowers, and under what conditions ultrapowers embed into corresponding cumulative powers or direct powers. We further offer an application of the framework to show a straightforward way of constructing a surreal field.
Paper Structure (8 sections, 90 theorems)

This paper contains 8 sections, 90 theorems.

Table of Contents

  1. Introduction
  2. Conventions
  3. A hierarchy of cumulative powers
  4. A concrete example
  5. Preservation and properties of cumulative powers
  6. Direct powers and the hierarchy
  7. Ultrapowers and the hierarchy
  8. An application to real closed fields

Key Result

Corollary 2.27

Let $\varphi \in \mathcal{L}^\sigma$ be a sentence. Then:

Theorems & Definitions (171)

  • Corollary 2.27
  • Theorem 4.20
  • Theorem 4.31
  • Definition 2.1
  • Definition 2.2: Level of an element
  • Definition 2.3: Cumulative power hierarchy
  • Definition 2.4: $\sigma$-structure of the cumulative power hierarchy and hereditary constant
  • Proposition 2.6
  • proof
  • Corollary 2.7
  • ...and 161 more