I don't believe that the set of all legal chess positions, with \$$\leq\$$ meaning "can produce by legal moves", is a preorder set, since it does not satisfy the reflexivity condition. For example, if the current position \$$x\$$ is a check in which the only legal response is a pawn move, then \$$x\nleq x\$$. (Maybe this can be fixed by allowing the number of intervening legal moves to be 0, and perhaps that's what you meant anyway.)