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To avoid creating new binomial posets by taking two of them with the same factorial function (i.e., number of maximal chains in an $n$-interval) and identifying their least elements, we should add the extra condition that there exists a maximal chain $\hat{0}< x _0< x _1< \cdots$ such that every element $x$ satisfies $x\leq x_n$ for some $n$. Backelin has constructed uncountably an uncountable number of nonisomorphic such binomial posets, all with the same factorial function. Most of these have the property that they contain two nonisomorphic intervals of the same length. See http://front.math.ucdavis.edu/0508.5397.
To avoid creating new binomial posets by taking two of them with the same factorial function (i.e., number of maximal chains in an $n$-interval) and identifying their least elements, we should add the extra condition that there exists a maximal chain $\hat{0}< x _0< x _1< \cdots$ such that every element $x$ satisfies $x\leq x_n$ for some $n$. Backelin has constructed uncountably nonisomorphic such binomial posets, all with the same factorial function. Most of these have the property that they contain two nonisomorphic intervals of the same length. See http://front.math.ucdavis.edu/0508.5397.