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We show that each $Q$-polynomial structure for $\\G$ yields a certain linear dependency among $RL^2$, $LRL$, $L^2R$, $L$. Define a partial order $\\le$ on $X$ as follows. For $y,z \\in X$ let $y \\le z$ whenever $\\partial(x,y)+\\partial(y,z)=\\partial(x,z)$, where $\\partial$ denotes path-length distance. We determine whether the above linear dependency gives this poset a uniform or strongly uniform structure. 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Fix $x \\in X$ and let $L$ (resp. $R$) denote the corresponding lowering (resp. raising) matrix. We show that each $Q$-polynomial structure for $\\G$ yields a certain linear dependency among $RL^2$, $LRL$, $L^2R$, $L$. Define a partial order $\\le$ on $X$ as follows. For $y,z \\in X$ let $y \\le z$ whenever $\\partial(x,y)+\\partial(y,z)=\\partial(x,z)$, where $\\partial$ denotes path-length distance. We determine whether the above linear dependency gives this poset a uniform or strongly uniform structure. 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