$\mathcal{L}\mathcal{R}$-Ending partisan rulesets

TL;DR

This paper explores a specific class of combinatorial game theory called $\\mathcal{L}\mathcal{R}$-ending partisan rulesets, analyzing their algebraic structures and providing examples to facilitate their analysis.

Contribution

It introduces the concept of $\\mathcal{L}\mathcal{R}$-ending partisan rulesets, develops their algebraic framework, and demonstrates how to analyze such rulesets with examples.

Findings

Defined algebraic structures of $\\mathcal{L}\mathcal{R}$-ending partisan rulesets

Introduced notation for positions in these rulesets

Provided examples illustrating analysis methods

Abstract

In this paper, we consider $\mathcal{L}\mathcal{R}$-ending partisan rulesets as a branch of combinatorial game theory. In these rulesets, the sets of options of both players are the same. However, there are two kinds of terminal positions. If the game ends in one of the terminal positions, then a player wins and if the game ends in the other terminal position, the other player wins. We introduce notations for positions in $\mathcal{L}\mathcal{R}$-ending partisan rulesets and show their algebraic structures. We also introduce some examples of $\mathcal{L}\mathcal{R}$-partisan rulesets and show how our results can be used for analyzing the rulesets.