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:
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.