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Asymptotic behavior of functions, asymptotic series and related topics

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How many upper sets in this decomposition of finite posets

Let $X$ be a finite poset. If $$X = X_1 \cup X_2$$ where $X_1$ and $X_2$ are strict upper sets, then a lot of properties of $X$ can be inferred from the smaller posets $X_1, X_2$ and $X_1\cap X_2$ (s …