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Let $(P,\leq)$ denote a poset and suppose $X \subseteq P$. Then the minimum element of the set of all upper bounds of $X$ can be denoted $\operatorname{sup} X$, or $\bigvee X$. Is there a similar notation for the collection of all minimal elements of the set of all upper bounds?

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There is none, as far as I know.

However, the set of all upper bound of a subset $X$ is sometimes denoted by $X^\uparrow$ and the set of all minimal elements of a subset $Y$ is sometimes denoted by $\min(Y)$; this gives us the notation $\min(X^\uparrow)$.

I think this is a usable notation, because if one wants to prove something about the set in question, the proof will probably use the set $X^\uparrow$. So it probably did not make sense to invent an "irreducible" notation.

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