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Exploring the Cube of Kleene Algebras and Multirelations: A New Perspective

This research delves into the intriguing intersection of multirelations and Kleene algebras, presenting a comprehensive analysis of their relationship. We define multirelations as subsets of AP(A) and explore their applications, particularly in the context of game theory. The study outlines our main results, provides detailed discussions, and sets the stage for future research directions. Our findings highlight the potential of multirelations in analyzing game strategies, contributing to the broader field of algebraic structures in mathematics.

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Exploring the Cube of Kleene Algebras and Multirelations: A New Perspective

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    1. The cube of Kleene algebras and the triangular prism of multirelations Koki Nishizawa (Tohoku Univ.) With Norihiro Tsumagari (Kagoshima Univ.) Hitoshi Furusawa (Kagoshima Univ.)

    2. Contents Background Overview of our main result Details of the result Future work

    3. Background

    4. Multirelation Def. A multirelation on A is a subset of AP(A). (P(A) is the power set of A) An ordinary binary relation on A is a subset of AA In this talk, I will give the relationship between multirelations and Kleene algebras. An ordinary on the other hand, a multirelationIn this talk, I will give the relationship between multirelations and Kleene algebras. An ordinary on the other hand, a multirelation

    5. e.g. Multirelation for game [Venema 03] Given A the states of a game board P?AA possible transition by player P Q?AA possible transition by player Q W?A the winnning sets of player P Def. Multirelation R for player P R={(a,X) | ?b. aPb and ?c. (bQc ? c?X)} Prop. If (a,W)?R, then player P can win at the next turn of a) (a,W)?R+RR+RRR+? Iteration of multirelation ? Multirelation is used to analyse games. Multirelation is used to analyse games.

    6. Kleene algebra (KA) algebraic structure (K,+,0,,1,*) for regular languages e.g. K the binary relations on A R* is the reflexive transitive closure of R It is used to represent properties of the reflexsive transitive closure of a multirelation.

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