Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Invariants of structures

Charlotte Aten

Abstract

We give a categorification of the notion of a mathematical structure originally given by Bourbaki in their set theory textbook. We show that any isomorphism-invariant property of a finite structure can be computed by counting the number of isomorphic copies of small substructures it contains. Our main theorem in this direction is a generalization of the classical result of Hilbert about elementary symmetric polynomials generating the algebra of all symmetric polynomials. We also show that, for structures built from sets, the Yoneda functor extends to a canonical embedding of any such category of structures into an associated category of structures in the sense of classical model theory.

Invariants of structures

Abstract

We give a categorification of the notion of a mathematical structure originally given by Bourbaki in their set theory textbook. We show that any isomorphism-invariant property of a finite structure can be computed by counting the number of isomorphic copies of small substructures it contains. Our main theorem in this direction is a generalization of the classical result of Hilbert about elementary symmetric polynomials generating the algebra of all symmetric polynomials. We also show that, for structures built from sets, the Yoneda functor extends to a canonical embedding of any such category of structures into an associated category of structures in the sense of classical model theory.
Paper Structure (17 sections, 18 theorems, 72 equations, 2 figures)

This paper contains 17 sections, 18 theorems, 72 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Structures, naïvely
  3. Substructures
  4. Finite structures
  5. Symmetric polynomials
  6. Definitions
  7. A generating set
  8. Example: domain digraphs
  9. Logic by counting
  10. Structures, formally
  11. Definition of a structure
  12. Parts of a structure
  13. Categories of structures
  14. Example: Pairs
  15. Example: Hypergraphs
  16. ...and 2 more sections

Key Result

Lemma 1

Given $y_{\mathbf{A}_1},\dots,y_{\mathbf{A}_k}\in Y^\rho_A$ we have that where $\mu\in\mathop{\mathrm{Pol}}\nolimits^\rho_A$.

Figures (2)

  • Figure 1: Composition is isotone
  • Figure 2: Morphism composition

Theorems & Definitions (102)

  • Definition 1: Substructure
  • Definition 2: Finite signature
  • Definition 3: Finite structure
  • Definition 4: Finite kinship class
  • Definition 5: Variables $X^\rho_A$
  • Definition 6: Monomial $y_{\mathbf{A}}$
  • Definition 7: Monomials $Y^\rho_A$
  • Definition 8: $(\rho,A)$ polynomial algebra
  • Lemma 1
  • proof
  • ...and 92 more