Isomorphism theorems for impartial combinatorial games

Mikhail Baltushkin; Dana C. Ernst; Nándor Sieben

doi:10.47443/dml.2025.071

Isomorphism theorems for impartial combinatorial games

Mikhail Baltushkin
, Dana C. Ernst
, Nándor Sieben

Mathematics and Statistics

Research output: Contribution to journal › Article › peer-review

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.

Original language	English (US)
Pages (from-to)	59-66
Number of pages	8
Journal	Discrete Mathematics Letters
Volume	16
DOIs	https://doi.org/10.47443/dml.2025.071
State	Published - 2025

Keywords

congruence relation
impartial game
isomorphism theorems
minimum quotient
option-preserving map

ASJC Scopus subject areas

Discrete Mathematics and Combinatorics

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10.47443/dml.2025.071

Cite this

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AB - 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.

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KW - impartial game

KW - isomorphism theorems

KW - minimum quotient

KW - option-preserving map

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JO - Discrete Mathematics Letters

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Isomorphism theorems for impartial combinatorial games

Abstract

Keywords

ASJC Scopus subject areas

Access to Document

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