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Isomorphism Theorems for Impartial Combinatorial Games

Mikhail Baltushkin, Dana C. Ernst, Nándor Sieben

TL;DR

Isomorphism theorems from universal algebra are extended to the category of optiongraphs (OGph) with option-preserving maps, providing a unified algebraic framework for impartial combinatorial games. The authors define quotient optiongraphs, including a minimum quotient, via congruence relations and prove the First through Fourth Isomorphism Theorems hold in this setting, preserving outcomes, remoteness, and extended nim-values. They also show how subcategories (rulegraphs and infinite-play optiongraphs) inherit these theorems, study simplicity of quotients, and analyze sums of optiongraphs and their quotients. The results yield practical, constructive tools for simplifying game analysis and comparing sums through principled quotienting.

Abstract

We introduce the category of optiongraphs and option-preserving maps as a model to study impartial combinatorial games. Outcomes, remoteness, and extended nim-values are preserved under option-preserving maps. We show that the four isomorphism theorems from universal algebra are valid in this category. Quotient optiongraphs, including the minimum quotient, provide simplifications that can help in the analysis of games.

Isomorphism Theorems for Impartial Combinatorial Games

TL;DR

Isomorphism theorems from universal algebra are extended to the category of optiongraphs (OGph) with option-preserving maps, providing a unified algebraic framework for impartial combinatorial games. The authors define quotient optiongraphs, including a minimum quotient, via congruence relations and prove the First through Fourth Isomorphism Theorems hold in this setting, preserving outcomes, remoteness, and extended nim-values. They also show how subcategories (rulegraphs and infinite-play optiongraphs) inherit these theorems, study simplicity of quotients, and analyze sums of optiongraphs and their quotients. The results yield practical, constructive tools for simplifying game analysis and comparing sums through principled quotienting.

Abstract

We introduce the category of optiongraphs and option-preserving maps as a model to study impartial combinatorial games. Outcomes, remoteness, and extended nim-values are preserved under option-preserving maps. We show that the four isomorphism theorems from universal algebra are valid in this category. Quotient optiongraphs, including the minimum quotient, provide simplifications that can help in the analysis of games.
Paper Structure (13 sections, 20 theorems, 12 equations, 10 figures)

This paper contains 13 sections, 20 theorems, 12 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Partitions of Optiongraphs
  4. Finite-Infinite-Mixed partition
  5. Quotient optiongraphs
  6. First Isomorphism Theorem
  7. Second Isomorphism Theorem
  8. Third Isomorphism Theorem
  9. Fourth Isomorphism Theorem
  10. Applications
  11. Subcategories
  12. Simplicity
  13. Sums

Key Result

Proposition 3.1

If $f:\mathsf{C}\to\mathsf{D}$ is an option-preserving surjection, then $f(F_\mathsf{C})=F_\mathsf{D}$, $f(I_\mathsf{C})=I_\mathsf{D}$, and $f(M_\mathsf{C})=M_\mathsf{D}$.

Figures (10)

  • Figure 1: Lattice of congruence relations of an optiongraph.
  • Figure 2: An optiongraph $\mathsf{D}$, its minimum quotient $\mathsf{D}/{\bowtie}$, together with extended nim-values and remoteness on $\mathsf{D}/{\bowtie}$.
  • Figure 3: First Isomorphism Theorem example.
  • Figure 4: Second Isomorphism Theorem example.
  • Figure 5: Third Isomorphism Theorem example for a rulegraph.
  • ...and 5 more figures

Theorems & Definitions (48)

  • Proposition 3.1
  • proof
  • Lemma 3.2
  • proof
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • proof
  • Example 3.5
  • Example 3.6
  • ...and 38 more