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I have been contemplating for some time if one can give the movements of chess pieces on a chess board an algebraic structure. I will give my ideas so far and hopefully someone can help me complete it.

To begin, here is a picture of a chessboard:

enter image description here

In the set which I put the algebra over, would be the set of moves themself. I am not sure how many operations should be given over the set, but till now I have been considering only one which is the composition of moves.

So, let me give an example. The bishop b1 to c3 could be represented as a move $\phi$ and pawn $c2$ to $c3$ could be represented as another move $\chi$. Now, in my set, the element $\phi$ followed by $\chi$ or $\chi$ followed by $\phi$ denoted as $\chi \phi$ and $\phi \chi$ does not exist because more than one piece can't be in a certain spot of the chess board. Essentially multiplication means doing one move after another.

However if Bishop g1 to f3 is denoted by $\alpha$ then $\chi \alpha$ and $\alpha \chi$ is in my set because they don't cause the two pieces occupying one spot issues.

In essence, closed under certain multiplications but not others.

It seems so generally that I don't have associativity or commutativity.

One nice property is that except for moves regarding pawns, everything in the set is actually invertible.

Is there anything more which can be said in this direction or is there another direction which gives a meaningful algebraic structure on the chess board? Thanks.


Note: At the moment I am thinking of free movement on the Chess Board. I am not accounting for the turn based system which is there when two people play. So, for instance black can move two turns in a row (legal to multiply black moves). However, if there is interesting results which occur from restricting it into only alternating colors moving, I'd be interested in that too.

The main restriction here is that I still keep the movement rules for pieces and between pieces. For example Bishop can only move in Ls, pawns one step at a time ( unless unmoved then two) etc. Also that of one chess square being occupied by only one piece.

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    $\begingroup$ Any game like chess has a game tree, which is a graph whose vertices are game positions and edges are moves passing between positions (en.wikipedia.org/wiki/Game_tree). This graph generates a free category, in the sense of category theory (en.wikipedia.org/wiki/Category_theory), whose morphisms are $k$-tuples of consecutive moves. I don't know that this algebraic structure is really any more useful than just the graph. $\endgroup$ Sep 19, 2022 at 22:06
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    $\begingroup$ Not sure if this applies to your purpose, but be aware the state of a chess game depends on more than the positions of pieces on the board and whose turn it is: Whether or not a king or rook has ever moved affects castling rules. The immediately previous move affects the en passant capture rule. Rules about stalemate involve more of the game's history. $\endgroup$
    – aschepler
    Sep 19, 2022 at 22:06
  • $\begingroup$ The word "algebra" here is vague. Usually, when talking about an algebra, we are not talking about operations that can be performed some times, but not others. It will certainly be difficult to use methods of the field of algebra to study it. $\endgroup$ Sep 19, 2022 at 22:10
  • $\begingroup$ If anything, it seems more like category theory. But then c2-c3 would be a different operation/map from each position. $\endgroup$ Sep 19, 2022 at 22:14
  • $\begingroup$ I think I have circumvented that problem by the choice of operation I am using. Aren't there any algebras studying "sometimes closed" operations?@ThomasAndrews $\endgroup$ Sep 19, 2022 at 22:16

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Hint:

... Much has been said of the affinity between mathematics and chess: two domains of human thought where very limited sets of rules yield inexhaustible depths, challenges, frustrations and beauty. ...


These introductory words are the starting point of the article The Mathematical Knight by Noam D. Elkies and Richard P. Stanley. It presents a nice collection of mathematical problems and approaches around chess.

One of the themes presented in this paper is combinatorial game theory. The paper refers also to On numbers and endgames: Combinatorial game theory in chess endgames by Noam D. Elkies which shows some notation and techniques which are used to mathematically analyse certain endgames. This paper does not use the algebraic notation you might have in mind, but it may provide you with some ideas, how certain chess positions and endgames can be described.

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    $\begingroup$ Thanks. That's a nice help $\endgroup$ Sep 20, 2022 at 17:37

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