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Categories of impartial rulegraphs and gamegraphs

Author

Listed:
  • Bojan Bašić

    (University of Novi Sad)

  • Paul Ellis

    (Rutgers University)

  • Dana C. Ernst

    (Northern Arizona University)

  • Danijela Popović

    (Mathematical Institute of the Serbian Academy of Sciences and Arts)

  • Nándor Sieben

    (Northern Arizona University)

Abstract

The traditional mathematical model for an impartial combinatorial game is defined recursively as a set of the options of the game, where the options are games themselves. We propose a model called gamegraph, together with its generalization rulegraph, based on the natural description of a game as a digraph where the vertices are positions and the arrows represent possible moves. Such digraphs form a category where the morphisms are option preserving maps. We study several versions of this category. Our development includes congruence relations, quotients, and isomorphism theorems and is analogous to the corresponding notions in universal algebra. The quotient by the maximum congruence relation produces an object that is essentially equivalent to the traditional model. After the development of the general theory, we count the number of non-isomorphic gamegraphs and rulegraphs by formal birthday and the number of positions.

Suggested Citation

  • Bojan Bašić & Paul Ellis & Dana C. Ernst & Danijela Popović & Nándor Sieben, 2024. "Categories of impartial rulegraphs and gamegraphs," International Journal of Game Theory, Springer;Game Theory Society, vol. 53(4), pages 1407-1433, December.
    • Handle: RePEc:spr:jogath:v:53:y:2024:i:4:d:10.1007_s00182-024-00921-3
    DOI: 10.1007/s00182-024-00921-3
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