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A Knaster--Reichbach type theorem for graph structures

Wiesław Kubiś, Andrzej Kucharski, Sławomir Turek

Abstract

We study the properties of a generic object $\mathbb{P}$ in the category of finite graphs. It turns out that this object, being topologically a Cantor set, has the Knaster--Reichbach type property. Namely, every homeomorphism and isomorphism $h\colon K\to L$ where $K$ and $L$ are nowhere dense closed sets in $\mathbb{P}$ and consisting only of isolated vertices in $K$ and $L$ can be extended to the autohomeomorphism and autoisomorphism of the whole graph $\mathbb{P}$.

A Knaster--Reichbach type theorem for graph structures

Abstract

We study the properties of a generic object in the category of finite graphs. It turns out that this object, being topologically a Cantor set, has the Knaster--Reichbach type property. Namely, every homeomorphism and isomorphism where and are nowhere dense closed sets in and consisting only of isolated vertices in and can be extended to the autohomeomorphism and autoisomorphism of the whole graph .
Paper Structure (3 sections, 11 theorems, 10 equations)

This paper contains 3 sections, 11 theorems, 10 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Categories of finite graphs

Key Result

Theorem 2.1

Assume that $\mathfrak{K}$ is a directed category with the amalgamation property and countable (up to isomorphism). Then $\mathfrak{K}$ has an Fraïssé sequence.

Theorems & Definitions (19)

  • Theorem 2.1: cf. KubFra Corollary 3.8
  • Theorem 2.2: KKT25
  • Theorem 3.1: cf. Camerlo Camerlo
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • Proposition 3.5
  • ...and 9 more