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Non-Archimedean Analogue of Chase's Lemma

Tomoki Mihara

Abstract

We formulate and verify a non-Archimedean counterpart of Chase's lemma. Following the framework by K.\ Eda removing restriction of cardinality from analogy on direct product between countable cardinal and non-$ω_1$-measurable cardinal, we extend the non-Archimedean counterpart of Chase's lemma to a non-Archimedean counterpart of the extension by K.\ Eda of the extension by M.\ Dugas and B.\ Zimmermann-Huisgen of Chase's lemma.

Non-Archimedean Analogue of Chase's Lemma

Abstract

We formulate and verify a non-Archimedean counterpart of Chase's lemma. Following the framework by K.\ Eda removing restriction of cardinality from analogy on direct product between countable cardinal and non--measurable cardinal, we extend the non-Archimedean counterpart of Chase's lemma to a non-Archimedean counterpart of the extension by K.\ Eda of the extension by M.\ Dugas and B.\ Zimmermann-Huisgen of Chase's lemma.

Paper Structure

This paper contains 3 sections, 14 theorems, 35 equations.

Table of Contents

  1. Introduction
  2. Convention
  3. Chase--Dugas--Zimmermann-Huisgen--Eda Theorem

Key Result

Theorem 1

For any countable set $I$, the canonical morphism $\mathbb{Z}^{\oplus} \to (\mathbb{Z}^I)^{\vee}$ is an isomorphism, and hence $\mathbb{Z}^{\oplus I}$ and $\mathbb{Z}^I$ are reflexive.

Theorems & Definitions (23)

  • Theorem 1: Specker's theorem
  • Theorem 2: Łoś's theorem
  • Theorem 3: Eda's theorem
  • Theorem 4: Chase's lemma
  • Theorem 5: Dugas--Zimmermann-Huisgen's extension of Chase's lemma
  • Theorem 6: Eda's extension of Dugas--Zimmermann-Huisgen's extension of Chase's lemma
  • Theorem 2.1: non-Archimedean Eda's extension of Dugas--Zimmermann-Huisgen's extension of Chase's lemma
  • Definition 2.2
  • Lemma 2.3
  • proof
  • ...and 13 more