The Folded (2D+1)-cube and Its Uniform Posets

Let I denote the folded (2D + 1)-cube with vertex set X and diameter D ae 3. Fix x a X. We first define a partial order ae on X as follows. For y, z a X let y ae z whenever a,(x, y) + a,(y, z) = a,(x, z). Let R (resp. L) denote the raising matrix (resp. lowering matrix) of I. Next we show that there exists a certain linear dependency among RL (2), LRL,L (2) R and L for each given Q-polynomial structure of I. Finally, we determine whether the above linear dependency structure gives this poset a uniform structure or strongly uniform structure.