Even Kakutani equivalence via (alpha)over-right-arrow and (beta)over-right-arrow equivalence in Z(2)

We show that an even Kakutani equivalence class of Z(2) actions is "spanned " by <(<alpha>)over right arrow> and <(<beta>)over right arrow> equivalence classes where <(<alpha>)over right arrow> = {1 + alpha (1), 1 + alpha (2)} , <(<beta>)over right arrow> = {1 + beta (1), 1 + beta (2)} and {1, alpha ( -1)(i), beta (-1)(i)} are rationally independent for i = 1, 2. Namely, give n such vectors <(<alpha>)over right arrow> and <(<beta>)over right arrow> a nd two evenly Kakutani equivalent Z(2) actions S and T, we show that U is < (<alpha>)over right arrow> -equivalent to S and <(<beta>)over right arrow> -equivalent to T.