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.)