What if we considered each arrangement of pieces on the board to be an element in the set of all configurations, and somehow, each new possible move to be sort of a transformation of the current state of the board into a new state?
Isn’t that like transformation groups where the group is made of possible “actions” on a geometric figure (such as those preserving symmetry)?
I don’t know what algebraic structure it has, but I’m considering that it could have certain algebraic properties, like possibly closure. Might we find generator elements or cycles within the object? Could we identify any interesting board configurations with special algebraic properties, or strategize to move towards a certain outcome based on what we know based on the algebraic properties takes us there?
We can try to frame its algebraic structure in other ways, I don’t know, like board “addition” or something where the change of state of the board is considered the addition of one board state (the empty board but with one piece) plus the current board stated.
The idea is that although there are so many combinatorial possibilities of board states, surely this is some kind of known, infinite combinatorial group or other algebraic object?
There are rules which came from group theory which allowed people to solve Rubik’s cubes, so could there be any parallel to Go?
Combinatorial game theory has some things to say about go, especially the late endgame. You want to check out
Berlekamp and Wolfe were able to analyze late endgames in enough detail using the notion of a sum of two games that they could construct and solve endgame puzzles that stumped 9-dan professionals. Sensei's Library gives an example from the book. White to move and win:
However, I should clarify that as far as I know this isn't really relevant to actually playing go in practice; that is, I don't think studying any of these books will help you actually win games.